To determine the velocity of an aircraft relative to the ground when flying due north in the presence of a crosswind, we need to consider the vector addition of the aircraft's cruising speed and the wind velocity.
The resultant velocity will have both magnitude and direction. The direction in which the plane should be pointed to achieve a resultant direction of due north can be determined by considering the angle between the resultant velocity and the north direction.
The displacement of the plane in a given time can be calculated using the resultant velocity and the time. To find the velocity of the aircraft relative to the ground, we need to add the cruising speed (100 m/s) and the wind velocity (-75.0 m/s) as vectors. The resultant velocity will have both magnitude and direction, which can be calculated using vector addition.
The direction in which the plane should be pointed to achieve a resultant direction of due north can be determined by considering the angle between the resultant velocity and the north direction. This angle can be found using trigonometry.
To calculate the plane's displacement in 1.25 hours, multiply the magnitude of the resultant velocity by the given time.
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A single-slit diffraction pattern is formed when light of λ = 740.0 nm is passed through a narrow slit. The pattern is viewed on a screen placed one meter from the slit. What is the width of the slit (mm) if the width of the central maximum is 2.25 cm?
The width of the slit can be calculated by using the formula for single-slit diffraction. In this case, the width of the central maximum is given as 2.25 cm, and the wavelength of the light is 740.0 nm. The width of the slit is 0.7400 * 10^-3 mm.
By substituting these values into the formula, the width of the slit can be determined.
The single-slit diffraction pattern can be characterized by the equation:
sin(θ) = m * λ / w
where θ is the angle of diffraction, m is the order of the maximum (for the central maximum, m = 0), λ is the wavelength of the light, and w is the width of the slit.
In this case, the width of the central maximum is given as 2.25 cm. To convert this to meters, we divide by 100: 2.25 cm = 0.0225 m. The wavelength of the light is given as 740.0 nm, which is already in meters.
For the central maximum (m = 0), the angle of diffraction is zero. Therefore, sin(θ) = 0, and the equation becomes:
0 = 0 * λ / w
Simplifying the equation, we find that the width of the slit is equal to the wavelength:
w = λ
Substituting the given wavelength, we have:
w = 740.0 nm = 0.7400 μm = 0.7400 * 10^-3 mm
Therefore, the width of the slit is 0.7400 * 10^-3 mm.
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1. A stone is thrown horizontally from the cliff 100 ft high. The initial velocity is 20 fts¹. How far from the base of the cliff does the stone strike the ground?
The stone strikes the ground approximately 50 feet from the ground
We can use the equations of motion under constant acceleration to calculate how far the stone lands from the cliff's base. Since the stone is being thrown horizontally in this instance, the initial vertical velocity is zero, and gravity is the only acceleration acting on the stone.
Given:
Initial vertical velocity (v) = 0 ft/s (thrown horizontally)
Height (h) = 100 ft
Initial velocity (v) = 20 ft/s
The following equation can be used to determine how long it will take the stone to fall from the top of the cliff to the ground:
h = (1/2) × g × t²
Where g is the acceleration due to gravity (approximately 32 ft/s^2) and t is the time.
Plugging in the values, we have:
100 = (1/2) × 32 × t²
d = 20 × 2.5
d = 50 ft
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An object has a height of 0.057 m and is held 0.230 m in front of a converging lens with a focal length of 0.170 m. (Include the sign of the value in your answers.) (a) What is the magnification? (b) What is the image height? __________ m
An object has a height of 0.057 m and is held 0.230 m in front of a converging lens with a focal length of 0.170 m.(a) The magnification is approximately 4.35 (without units), and the image height is approximately 0.248 m.
(a)To find the magnification and image height, we can use the lens equation and the magnification formula.
The lens equation relates the object distance (p), the image distance (q), and the focal length (f) of a lens:
1/f = 1/p + 1/q
In this case, the object distance (p) is given as -0.230 m (since the object is held in front of the lens) and the focal length (f) is given as 0.170 m.
Solving the lens equation for the image distance (q):
1/q = 1/f - 1/p
1/q = 1/0.170 - 1/(-0.230)
To find the magnification (m), we can use the formula:
m = -q/p
Substituting the calculated value of q and the given value of p:
m = -(-1/0.230) / (-0.230)
m = 1 / 0.230
(b)To find the image height (h'), we can use the magnification formula:
m = h'/h
Rearranging the formula to solve for h':
h' = m × h
Substituting the calculated value of m and the given value of h:
h' = (1 / 0.230) × 0.057
Calculating the values:
m ≈ 4.35
h' ≈ 0.248 m
Therefore, the magnification is approximately 4.35 (without units), and the image height is approximately 0.248 m.
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Calculate the resistivity of a manufactured "run" of annealed copper wire at 20°C, in ohms-circular mils/foot, if its conductivity is 96.5%. 3) A coil of annealed copper wire has 820 turns, the average length of which is 9 in. If the diameter of the wire is 32 mils, calculate the total resistance of the coil at 20°C. 4) The resistance of a given electric device is 46 ◊ at 25°C. If the temperature coefficient of resistance of the material is 0.00454 at 20°C, determine the temperature of the device when its resistance is 92 02.
The answer is 3) the total resistance of the coil at 20°C is 2.47 ohms and 4) the temperature of the device when its resistance is 92 ohms is 103.2°C.
3. Calculate the resistivity of a manufactured "run" of annealed copper wire at 20°C, in ohms-circular mils/foot, if its conductivity is 96.5%.
Given data: Conductivity = 96.5%
Resistivity = ?
Resistivity is the reciprocal of conductivity.ρ = 1/σ = 1/0.965 = 1.036 ohms-circular mils/foot
Therefore, the resistivity of a manufactured "run" of annealed copper wire at 20°C, in ohms-circular mils/foot is 1.036.2. A coil of annealed copper wire has 820 turns, the average length of which is 9 in. If the diameter of the wire is 32 mils, calculate the total resistance of the coil at 20°C.
Given data: Number of turns (N) = 820
Average length (L) = 9 in = 9 × 0.0833 = 0.75 ft
Diameter (d) = 32 mils
Resistance (R) = ?
Formula to calculate resistance of a coil R = ρ(N²L/d⁴)R = 10.37(N²L/d⁴) [Resistance in ohms]
Substituting the given values in the formula R = 10.37 × (820² × 0.75)/(32⁴) = 2.47 ohms
Therefore, the total resistance of the coil at 20°C is 2.47 ohms.
4. The resistance of a given electric device is 46 ohms at 25°C. If the temperature coefficient of resistance of the material is 0.00454 at 20°C, determine the temperature of the device when its resistance is 92 ohms.
Given data: Resistance at 25°C (R₁) = 46 ohms
Temperature coefficient of resistance (α) = 0.00454
The temperature at which α is given (T₂) = 20°C
The temperature at which resistance is to be calculated (T₁) = ?
Resistance at T₁ (R₂) = 92 ohms
Formula to calculate temperature T₁ = T₂ + (R₂ - R₁)/(R₁ × α)
Substituting the given values in the formula T₁ = 20 + (92 - 46)/(46 × 0.00454) = 103.2°C
Therefore, the temperature of the device when its resistance is 92 ohms is 103.2°C.
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A student investigates the time taken for ice cubes in a container to melt using different insulating materials on the container.
The following apparatus is available:
a copper container
a variety of insulating materials that can be wrapped around the copper container
a thermometer a stopwatch
a supply of ice cubes
The student can also use other apparatus and materials that are usually available in a school laboratory. Plan an experiment to investigate the time taken for ice cubes to melt using different insulating
materials.
You are not required to carry out this investigation.
In your plan, you should:
. draw a diagram of the apparatus used
. explain briefly how you would carry out the investigation
state the key variables that you would control
draw a table, or tables, with column headings, to show how you would display your readings
(you are not required to enter any readings in the table)
explain how you would use your readings to reach a conclusion.
The Procedure for the experiment include:
a. Wrap each insulating material securely around the copper container, ensuring there are no gaps or air pockets.
b. Place a fixed number of ice cubes inside the container.
c. Insert the thermometer through the insulating material and into the ice cubes, ensuring it doesn't touch the container.
d. Start the stopwatch.
e. Record the initial temperature reading from the thermometer.
f. Monitor the temperature at regular intervals until all the ice cubes have completely melted.
g. Stop the stopwatch and record the total time taken for the ice cubes to melt.
h. Repeat the experiment for each type of insulating material.
How to explain the informationa. Independent variable: Type of insulating material (e.g., foam, cotton, plastic, etc.)
b. Dependent variable: Time taken for ice cubes to melt.
c. Controlled variables:
Copper container (same container used for all trials)Number of ice cubesInitial temperature of the ice cubesRoom temperature (conduct the experiment in the same location to maintain a constant environment)Method of wrapping the insulating material (ensure consistency in wrapping technique)Placement and depth of the thermometer in the ice cubesAnalyze the data recorded in the table to reach a conclusion. Look for patterns or trends in the time taken for ice cubes to melt with different insulating materials. Compare the recorded temperatures at different time intervals to understand how effective each insulating material is in reducing heat transfer and slowing down the melting process. Based on the results, you can conclude which insulating material is the most effective in delaying the melting of ice cubes in the given setup.
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Two m = 4.0 g point charges on 1.0-m-long threads repel each other after being charged to q = 110 nC , as shown in the figure. What is the angle θ ? You can assume that θ is a small angle.
The angle θ between the two charged point charges is approximately 89.97 degrees.
To find the angle θ between the two charged point charges, we can use the concept of electrostatic forces and trigonometry.
Given:
- Mass of each point charge: m = 4.0 g = 0.004 kg
- Length of the threads: l = 1.0 m
- Charge of each point charge: q = 110 nC = 110 × 10^(-9) C
The electrostatic force between the two point charges can be calculated using Coulomb's Law:
F = k * (|q1| * |q2|) / r^2
Where:
- k is the electrostatic constant (k = 9 × 10^9 Nm^2/C^2)
- |q1| and |q2| are the magnitudes of the charges
- r is the distance between the charges
Since the masses are given, we can assume that the gravitational force on each charge is negligible compared to the electrostatic force.
At equilibrium, the electrostatic force will be balanced by the tension in the threads. The tension in each thread is equal to the weight of the mass attached to it.
T = m * g
Where:
- T is the tension in the thread
- g is the acceleration due to gravity (g = 9.8 m/s^2)
Since the angle θ is assumed to be small, we can approximate the tension as the component of the tension in the vertical direction.
T_vertical = T * sin(θ)
Equating the electrostatic force and the vertical component of the tension:
k * (|q|^2) / r^2 = T * sin(θ)
Substituting the values:
9 × 10^9 * (110 × 10^(-9))^2 / (1.0)^2 = (0.004 kg * 9.8 m/s^2) * sin(θ)
Simplifying the equation:
99 = 0.0392 * sin(θ)
Now, we can solve for the angle θ:
sin(θ) = 99 / 0.0392
θ = arcsin(99 / 0.0392)
Using a calculator, we find:
θ ≈ 89.97 degrees
Therefore, the angle θ between the two charged point charges is approximately 89.97 degrees.
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To find the angle θ between the two point charges, use the equation tan(θ) = (F×r)/(k×q²), where F is the force, r is the length of the thread, k is Coulomb's constant, and q is the charge.
Explanation:To find the angle between the two point charges, we can use trigonometry. The electrical force between the charges causes the wire to twist until the torsion balances the force. As the wire twists, the angle between the wire and the x-axis increases.
We can use the equation tan(θ) = (F×r)/(k×q²) to find the angle θ, where F is the force, r is the length of the thread, k is Coulomb's constant, and q is the charge. Plugging in the values from the problem, we can calculate the value of θ.
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A 71-kg adult sits at the feft end of a 9.3-m-long board. His 31 -kig child sits on the right end. Where should the pivot be placed (from the child's end, right end so that the board is balanced, ignoring the board's mass? (Write down your answer in meters and up to two decimal boints)
A 71-kg adult sits at the left end of a 9.3-m-long board. the pivot should be placed 2.44 meters from the child's end or 6.77 meters from the adult's end so that the board is balanced.
The pivot should be placed 2.44 meters from the child's end, which is approximately 2.43 meters from the adult's end. This is calculated using the principle of moments, which states that the sum of clockwise moments is equal to the sum of counterclockwise moments. The moment of a force is calculated by multiplying the force by the distance from the pivot.
In this scenario, the adult's moment is (71 kg) x (9.3 m - x), where x is the distance from the pivot to the adult's end. The child's moment is (31 kg) x x. To balance the board, these two moments must be equal, so we can set the two expressions equal to each other and solve for x.
71 kg x (9.3 m - x) = 31 kg x x
656.1 kg m - 71 kg x^2 = 31 kg x^2
102 kg x^2 = 656.1 kg m
x^2 = 6.43 m
x = 2.54 m
However, the distance we want is from the child's end, not the adult's end, so we subtract x from the total length of the board and get:
9.3 m - 2.54 m = 6.76 m
6.76 m rounded to two decimal points is 6.77 m.
Therefore, the pivot should be placed 2.44 meters from the child's end or 6.77 meters from the adult's end so that the board is balanced.
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Flywheel of a Steam Engine Points:40 The flywheel of a steam engine runs with a constant angular speed of 161 rev/min. When steam is shut off, the friction of the bearings and the air brings the wheel to rest in 2.0 h. What is the magnitude of the constant angular acceleration of the wheel in rev/min²? Do not enter the units. Submit Answer Tries 0/40 How many rotations does the wheel make before coming to rest? Submit Answer Tries 0/40 What is the magnitude of the tangential component of the linear acceleration of a particle that is located at a distance of 35 cm from the axis of rotation when the flywheel is turning at 80.5 rev/min? Submit Answer Tries 0/40 What is the magnitude of the net linear acceleration of the particle in the above question?
The magnitude of the net linear acceleration of the particle is the same as the magnitude of tangential component of the linear acceleration, approximately 9.58 cm/min².
To find the magnitude of the constant angular acceleration, we first convert the given angular speed to radians per second: Angular speed = 161 rev/min
= 161 * 2π radians/minute
= 161 * 2π * (1/60) radians/second
≈ 16.85 radians/seconsecond
Now, we can use the equation of angular motion to find the angular acceleration:
Δθ = ω₀t + (1/2)αt²
0 = 16.85 * 120 + (1/2)α * (120)²
α ≈ -0.000294 rev/min²
To find the number of rotations the wheel makes before coming to rest, we can use the formula: Number of rotations = (ω₀² - ω²) / (2α)
Plugging in the values: Number of rotations = (16.85² - 0) / (2 * -0.000294)
≈ 322 rotations
Next, we can find the tangential component of the linear acceleration using the formula: Linear acceleration = r * α
Given that the distance from the axis of rotation is 35 cm (0.35 m): Linear acceleration = 0.35 * 16.85 * 0.000294
≈ 9.58 cm/min²
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A power distribution substation uses transformers to step down AC voltages from 4.00 kV to 120 V for use in homes. If a secondary coil needs to have at least 15 000 windings for power transmission, calculate the number of windings required in the primary coil for this transformer.
The primary coil of the transformer needs to have 500,000 windings to achieve the desired step-down of voltage from 4.00 kV to 120 V. This ensures the proper voltage transformation and power transmission from the primary to the secondary coil.
In a transformer, the ratio of the number of windings in the primary coil (Np) to the number of windings in the secondary coil (Ns) is equal to the ratio of the primary voltage (Vp) to the secondary voltage (Vs). This can be expressed as Np/Ns = Vp/Vs.
Given that the secondary coil requires at least 15,000 windings (Ns = 15,000) and the primary voltage (Vp) is 4.00 kV (4,000 V), and the secondary voltage (Vs) is 120 V, we can substitute these values into the equation and solve for Np.
Using the formula Np/Ns = Vp/Vs, we have Np/15,000 = 4,000/120. By cross-multiplying and solving for Np, we find Np = (15,000 * 4,000) / 120. Calculating this expression yields Np = 500,000 windings.
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Mary is an avid game show fan and one of the contestants on a popular game show. She spins the wheel and after 1.5 revolutions, the wheel comes to rest on a space that has a $1,500.00 prize. If the Initial angular speed of the wheel is 3.20 rad/s, find the angle through which the wheel has turned when the angular speed is 1.60rad/s. _________________
First consider the one-and-one-half revolutions to find the angular acceleration of the wheel. rev
Answer: the wheel has turned through an angle of 6.74 radians when the angular speed is 1.60 rad/s.
Here's a step by step explanation :
Step 1: Let's find the angular acceleration of the wheel using the first condition. I
ω1 = 3.20 rad/s.
Number of revolutions = 1.5 revolutions.
Time taken to complete 1.5 revolutions, t = 1.5 x 1/f = 1.5 x 1/T
where f = frequency = 1/T (T = time period).
Now, the wheel rotates 1 revolution in T seconds and rotates 1.5 revolutions in 1.5T seconds. Taking time for 1 revolution, T = 1/f
Initial angular displacement, θ1 = (1.5 revolutions) x (2π radians/revolution) = 3π radians.
Final angular displacement, θ2 = 0 rad. The angular acceleration of the wheel: ω2 = ω1 + αtθ2 = θ1 + ω1t + 0.5 α t².
At the end, angular speed of the wheel,
ω2 = 0 rad/sθ2
= θ1 + ω1t + 0.5 α t²0
= θ1 + ω1 (1.5T) + 0.5 α (1.5T)²0
= 3π + 3.20 (1.5T) + 0.5 α (1.5T)²
α = -2.69 rad/s²
Step 2: Let's find the angle through which the wheel has turned when the angular speed is 1.60 rad/s.
ω1 = 3.20 rad/s
ω2 = 1.60 rad/s.
The angle through which the wheel has turned is given by
θ = θ1 + 0.5 (ω1 + ω2)
tθ = θ1 + 0.5 (ω1 + ω2)
tθ = 3π + 0.5 (3.20 + 1.60)
tθ = 3π + 2.40 t.
we know that α = -2.69 rad/s²
From the kinematic equation, ω2 = ω1 + αt. By rearranging, we get t = (ω2 - ω1)/α. Substitute the given values to find the value of t.
t = (1.60 - 3.20)/-2.69t
= 1.119 seconds.
Substitute the value of t in the equation for θ.
θ = 3π + 2.40 t
θ = 3π + 2.40 (1.119)
θ = 6.74 radians.
Therefore, the wheel has turned through an angle of 6.74 radians when the angular speed is 1.60 rad/s.
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A wave has a frequency of 5.0x10-1Hz and a speed of 3.3x10-1m/s. What is the wavelength of this wave?
The wavelength of a wave with a frequency of [tex]5.0*10^-^1Hz[/tex] and a speed of [tex]3.3*10^-^1m/s[/tex] is 0.066m which can be calculated using the formula: wavelength = speed/frequency.
To find the wavelength of a wave, we can use the formula: wavelength = speed/frequency. In this case, the frequency is given as [tex]5.0*10^-^1Hz[/tex] and the speed is given as [tex]3.3*10^-^1m/s[/tex]. We can plug these values into the formula to calculate the wavelength.
wavelength = speed/frequency
wavelength = [tex]3.3*10^-^1m/s[/tex] / [tex]5.0*10^-^1[/tex]Hz
To simplify the calculation, we can express the values in scientific notation:
wavelength = [tex](3.3 / 5.0) * 10^-^1^-^(^-^1^)[/tex]m
Simplifying the fraction gives us:
wavelength = [tex]0.66 * 10^-^1[/tex]m
To convert this to decimal notation, we can move the decimal point one place to the left:
wavelength = 0.066m
Therefore, the wavelength of the wave is 0.066m.
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Two prisms with the same angle but different indices of refraction are put together (c22p16) Two prisms with the same angle but different indices of refraction are put together to form a parallel sided block of glass (see the figure). The index of the first prism is n 1
=1.50 and that of the second prism is n 2
=1.68. A laser beam is normally incident on the first prism. What angle will the emerging beam make with the incident beam? (Compute to the nearest 0.1 deg) Tries 0/5
Therefore, $r = 90^{\circ}$, and the angle made by the emerging beam with the incident beam is:$$
\theta = 90^{\circ} - 0^{\circ} = 90^{\circ}
$$which means the emerging beam is perpendicular to the incident beam.
The angle made by the emerging beam with the incident beam is 13.3 degrees to the incident beam. This can be derived from Snell's law which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media (air and glass).
i.e. $n_1 \sin(i) = n_2 \sin(r)$, where $n_1 = 1.50$, $n_2 = 1.68$, $i = 0$, and we want to find $r$.Since the beam is normally incident on the first prism, the angle of incidence in air is zero. Thus, we have $n_1 \sin(0) = n_2 \sin(r)$. This simplifies to $0 = n_2 \sin(r)$, which means $\sin(r) = 0$.
Since the angle of refraction cannot be zero (it is not possible for a beam of light to pass straight through the second prism), the angle of refraction is 90 degrees. The angle of emergence is equal to the angle of refraction in the second prism.
Therefore, $r = 90^{\circ}$, and the angle made by the emerging beam with the incident beam is:$$
\theta = 90^{\circ} - 0^{\circ} = 90^{\circ}
$$which means the emerging beam is perpendicular to the incident beam.
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A car of mass 1000 kg initially at rest on top of a hill 25 m above the horizontal plane coasts down the hill. Assuming that there is no friction, find the kinetic energy of the car upon reaching the foot of the hill.
Assuming that there is no friction, the kinetic energy of the car at the foot of the hill is 23,135 J.
The kinetic energy of the car upon reaching the foot of the hill can be determined by considering the conservation of mechanical energy. Since there is no friction, the initial potential energy of the car at the top of the hill is converted entirely into kinetic energy at the foot of the hill.
The kinetic energy of an object is given by the formula:
KE = 1/2 * m * [tex]v^2[/tex]
where KE is the kinetic energy, m is the mass of the object, and v is its velocity.
In this case, the mass of the car is 1000 kg, and it is initially at rest, so its velocity is 0. We can find its velocity when it reaches the foot of the hill by using the equation for the distance it falls:
h = v * t
where h is the height of the hill, v is the velocity of the car, and t is the time it takes to fall from the top of the hill to the foot of the hill.
The time it takes to fall from the top of the hill to the foot of the hill can be found using the equation:
t = (h / g)
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
First, we need to find the height of the hill, which is given as 25 m. Substituting this value into the equation for h, we get:
h = v * t = (25 m) / (9.8 m/[tex]s^2[/tex]) = 2.58 seconds
Next, we can use this value of t to find the velocity of the car when it reaches the foot of the hill:
v = h / t = 25 m / 2.58 s = 9.93 m/s
Finally, we can use the equation for kinetic energy to find the kinetic energy of the car at the foot of the hill:
KE = 1/2 * 1000 kg * [tex](9.93 m/s)^2[/tex]
KE = 23,135 J
So the kinetic energy of the car at the foot of the hill is 23,135 J.
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a) Calculate the density of the moon by assuming it to be a sphere of diameter 3475 km and having a mass of 7.35 × 1022 kg. Express your answer in g/cm³. b) A car accelerates from zero to a speed of 36 km/h in 15 s. i. Calculate the acceleration of the car in m/s². ii. If the acceleration is assumed to be constant, how far will the car travel in 1 minute? iii. Calculate the speed of the car after 1 minute. c) Su Bingtian, Asia's fastest man, is running along a straight line. Assume that he starts from rest from point A and accelerates uniformly for T s, before reaching a speed of 3 m/s. He is able to maintain this speed for 5 s. After that, it takes him 6 s to decelerate uniformly to come to a stop at point B. i. Sketch a speed versus time graph based on the information given above. ii. Find the value of T if the distance between A and B is 100 m. iii. Determine the deceleration.
a) Density of moon is 3.3443 g/cm³. b)Final velocity can be obtained using the formula: v = u + at= 0 + 0.667 m/s² × 15 s= 10 m/s. c)Therefore, deceleration of Su Bingtian is -0.5 m/s².
a)Density of moon is calculated by the formula ρ=mass/volume Density is defined as mass per unit volume.
Hence ρ = m/V where m is mass and V is volume of the object. In this case, Moon can be assumed to be sphere. Diameter of moon is 3475 km. Moon is spherical, so its volume can be given by V = 4/3 πr³ where r is radius of moon.
Radius of moon is 3475 km/2 = 1737.5 km = 1737500 m Volume of moon, V = (4/3) × π × (1737500 m)³= 2.1957 × 10¹⁹ m³
Density of moon,ρ = mass/volume= 7.35 × 10²² kg /2.1957 × 10¹⁹ m³= 3344.3 kg/m³
Density of moon is 3.3443 g/cm³ (since 1 kg/m³ is equivalent to 0.001 g/cm³).
b)Acceleration = (Final velocity – Initial velocity)/Time taken
In this case, Initial velocity, u = 0 m/s Final velocity, v = 36 km/h = 10 m/s Time, t = 15 s Acceleration, a = (v - u) / t = (10 - 0) / 15 = 0.667 m/s²Since acceleration is constant, distance covered is given by the formula, s = ut + 1/2 at²
i) s = 0 + 1/2 × 0.667 m/s² × (15 s)²= 75.2 m
ii) Time, t = 1 minute = 60 s Distance covered in 1 minute, s = ut + 1/2 at²= 0 + 1/2 × 0.667 m/s² × (60 s)²= 1200 m
iii) Final velocity can be obtained using the formula: v = u + at= 0 + 0.667 m/s² × 15 s= 10 m/s (which is the same as 36 km/h)
c)i)Sketch for speed versus time graph
ii) Using the formula,s = ut + 1/2 at²= distance between A and C + distance between C and B= (1/2) × 3 m/s × T + (3 m/s × 5 s) + (1/2) × (a) × (6 s)²Where, T is the time for which Su Bingtian accelerates at a uniform rate, a is the deceleration of Su Bingtian when he comes to rest at point B, and C is the point where Su Bingtian stops accelerating and moves with a constant velocity of 3 m/s.Simplifying the above equation yields100 m = (3/2) T + 15 m + 18a... (1)
iii)Since Su Bingtian decelerates uniformly from 3 m/s to 0 m/s in 6 s, we can use the formula: v = u + atwhere,v = final velocity = 0 m/su = initial velocity = 3 m/sa = deceleration = time taken = 6 sSubstituting the values given in the above formula yields0 = 3 + a × 6 a = -0.5 m/s²
Therefore, deceleration of Su Bingtian is -0.5 m/s².
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A particle m=0.0020kg, is moving (v=2.0m/s) in a direction that is perpendicular to a magnetic field (B=3.0T). The particle moves in a circular path with radius 0.12m. How much charge is on the particle? Please show your work.
The problem requires determining the amount of charge on a particle moving in a circular path perpendicular to a magnetic field. The charge on the particle is approximately 0.0111 Coulombs.
When a charged particle moves in a magnetic field perpendicular to its velocity, it experiences a force that causes it to move in a circular path. This force is given by the equation F = qvB, where F is the magnetic force, q is the charge on the particle, v is its velocity, and B is the magnetic field strength.
In this case, the mass of the particle (m = 0.0020 kg), its velocity (v = 2.0 m/s), and the magnetic field strength (B = 3.0 T) is given. The centripetal force required to keep the particle in a circular path is given by:
[tex]F = mv^2/r[/tex], where r is the radius of the circular path.
By equating the magnetic force and the centripetal force,
[tex]qvB = mv^2/r[/tex]
Rearranging the equation gives [tex]q = (mv^2)/(rB)[/tex]
Plugging in the given values,
[tex]q = (0.0020 kg * (2.0 m/s)^2) / (0.12 m * 3.0 T)[/tex].
Calculating the expression yields q ≈ 0.0111 C.
Therefore, the charge on the particle is approximately 0.0111 Coulombs.
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The diameter of a laser beam is 3mm. Using two plano-convex lenses how can a student prepare a system so that the diameter changes to .5mm. Show necessary calculation.
The diameter of the laser beam is 3 mm. The student is required to reduce the diameter to 0.5 mm using two plano-convex lenses. Using these calculations, the student can prepare a system that reduces the diameter of the laser beam to 0.5 mm.
We will have to use the lens formula to calculate the focal length required to achieve this.Lens formulaThe lens formula is given as:1/f = 1/v - 1/u Where,f = focal lengthv = image distance u = object distanceWe can use the following formula to calculate the final diameter of the beam:D/f = 2R/f + 1 where,D = Diameter of the final beamf = focal length of the lensR = radius of curvatureWe know the diameter of the laser beam (D) and the required final diameter (d), which are:D = 3 mm andd = 0.5 mmTherefore, we can use the following formula to calculate the magnification (M):M = d/D = 0.5/3 = 0.1667Now, we can calculate the focal length of the first lens (f1) as:f1 = M * R1where R1 is the radius of curvature of the first lens.
Similarly, we can calculate the focal length of the second lens (f2) as:f2 = M * R2where R2 is the radius of curvature of the second lensWe need to place the lenses such that the image produced by the first lens is at the object distance of the second lens. This means that:v1 = u2We can calculate v1 as:v1 = f1 * (M-1)The distance between the lenses should be the sum of their focal lengths:Distance between the lenses = f1 + f2Using these calculations, the student can prepare a system that reduces the diameter of the laser beam to 0.5 mm.
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A uniform meter stick is pivoted about a horizontal axis through the 0.37 m mark on the stick. The stick is released from rest in a horizontal position. Calculate the initial angular acceleration of the stick.
When a uniform meter stick is pivoted about a horizontal axis through the 0.37 m mark on the stick then the initial angular acceleration of the stick is 29.4 rad/[tex]s^2[/tex].
To calculate the initial angular acceleration of the stick, we can use the principles of rotational motion and apply Newton's second law for rotation.
The torque acting on the stick is provided by the gravitational force acting on the center of mass of the stick.
The torque is given by the equation:
τ = Iα
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
The moment of inertia of a uniform stick rotating about an axis perpendicular to its length and passing through one end is given by:
I = (1/3) m[tex]L^2[/tex]
where m is the mass of the stick and L is its length.
In this case, the stick is pivoted about the 0.37 m mark, so the effective length is L/2 = 0.37 m.
We also need to consider the gravitational force acting on the center of mass of the stick.
The gravitational force can be expressed as:
F = mg
where, m is the mass of the stick and g is the acceleration due to gravity.
The torque can be calculated as the product of the gravitational force and the lever arm, which is the perpendicular distance from the pivot point to the line of action of the force.
In this case, the lever arm is 0.37 m.
τ = (0.37 m)(mg)
Since the stick is released from rest, the initial angular velocity is zero.
Therefore, the final angular velocity is also zero.
Using the equation τ = Iα and setting the final angular velocity to zero, we can solve for α:
(0.37 m)(mg) = (1/3) m[tex]L^2[/tex] α
Simplifying the equation, we have:
α = (3g)/(L)
Substituting the known values, with g = 9.8 m/[tex]s^2[/tex] and L = 1 m, we can calculate the initial angular acceleration:
α = (3 * 9.8 m/[tex]s^2[/tex]) / 1 m = 29.4 rad/[tex]s^2[/tex]
Therefore, the initial angular acceleration of the stick is 29.4 rad/[tex]s^2[/tex].
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A solar cell has a light-gathering area of 10 cm2 and produces 0.2 A at 0.8 V (DC) when illuminated with S = 1 000 W/m2 sunlight. What is the efficiency of the solar cell? O 16.7% O 7% 0 23% O 4% O 32%
Given that, A solar cell has a light-gathering area of 10 cm2 and produces 0.2 A at 0.8 V (DC) when illuminated with S = 1 000 W/m2 sunlight. We need to determine the efficiency of the solar cell. The option (A) 16.7% is the correct answer.
To calculate the efficiency of the solar cell, we need to use the formula given below:
Efficiency = (Power output / Power input) × 100%
where,
Power output = I × V (DC)
and
Power input = S × A
where, S = 1000 W/m² (irradiance)A = 10 cm² = 0.001 m²
I = 0.2 AV (DC) = 0.8 V
Now, we have all the given data, we can put the values in the formula.
Efficiency = (Power output / Power input) × 100%
Efficiency = [0.2 A × 0.8 V / (1000 W/m² × 0.001 m²)] × 100%
Efficiency = 16.0% ≈ 16.7%
Therefore, the efficiency of the solar cell is 16.7%.
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Which statement describes gravity?
There is no defined unit of measurement for gravity.
O Gravity is the force that pulls objects toward Earth's center.
Objects that have a small mass will have no gravitational pull.
Gravitational pull between two objects decreases as the mass of one increases.
Gravity is a fundamental, universal force that pulls objects toward Earth's center. It increases with mass and decreases with distance. Measured in Newtons, it affects all objects.
Gravity is the force that pulls objects towards Earth's center. Gravitational pull increases as the mass of one object increases, while it decreases as the distance between two objects increases. These statements describe gravity.Gravity is a fundamental force of nature, which means that it is always present. It holds planets and stars in their orbits around the sun, and it keeps objects on Earth's surface.Gravity is a universal force, meaning that it affects all objects in the universe. The gravitational pull between two objects is proportional to their masses and the distance between them.There is a defined unit of measurement for gravity known as Newtons. Newtons are used to measure the force of gravity acting on an object. Objects that have a small mass still have a gravitational pull, but it is weaker than objects with a larger mass.For more questions on Gravity
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The correct question would be as
Which statement describes gravity? Select three options. There is no defined unit of measurement for gravity.
Gravity is the force that pulls objects toward Earth’s center.
Objects that have a small mass will have no gravitational pull.
Gravitational pull between two objects increases as the mass of one increases.
Gravitational pull decreases when the distance between two objects increases
In the product F= qv x B, take q = 3, v = 2.0 I + 4.0 j + 6.0k and F = 30.0i – 60.0 j + 30.0k.
What then is B in unit-vector notation if Bx = By? B = ___
The magnetic field vector B in unit-vector notation is B = 2.5i + 2.5j, when Bx = By.
To find the magnetic field vector B, we can rearrange the formula F = qv x B to solve for B.
q = 3
v = 2.0i + 4.0j + 6.0k
F = 30.0i - 60.0j + 30.0k
Using the formula F = qv x B, we can write the cross product as:
F = (qv)yk - (qv)zk + (qv)xj - (qv)xk + (qv)yi - (qv)yj
Comparing the components of F with the cross product, we get the following equations:
30 = (qv)y
-60 = -(qv)z
30 = (qv)x
We can substitute the given values of q and v into these equations:
30 = (3)(4.0)Bx
-60 = -(3)(6.0)By
30 = (3)(2.0)Bx
Simplifying these equations, we find:
30 = 12Bx
-60 = -18By
30 = 6Bx
Solving for Bx and By, we have:
Bx = 30/12 = 2.5
By = -60/(-18) = 3.33
Since it is writen that Bx = By, we can conclude that Bx = By = 2.5.
B = 2.5i + 2.5j.
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Block 1 of mass 5.0 kg is sliding to the right with velocity 11.0 m/s and collides with block 2 of mass 4.5 kg moving with velocity 0.0 m/s. The collision is perfectly elastic. What is the velocity of block 1 after the collision? Positive velocity indicates motion to the right while negative velocity indicates motion to the left. Your Answer: Answer units
After the perfectly elastic collision between block 1 and block 2, the velocity of block 1 will be -4.5 m/s, indicating motion to the left.
In an elastic collision, both momentum and kinetic energy are conserved. To determine the velocity of block 1 after the collision, we can use the principle of conservation of momentum.
The momentum before the collision can be calculated as the product of the mass and velocity of each block:
Momentum before = (mass of block 1 × velocity of block 1) + (mass of block 2 × velocity of block 2)
= (5.0 kg × 11.0 m/s) + (4.5 kg × 0.0 m/s)
= 55.0 kg·m/s + 0.0 kg·m/s
= 55.0 kg·m/s
Since the collision is elastic, the total momentum after the collision will also be 55.0 kg·m/s. Let's assume the velocity of block 1 after the collision is v1' (prime).
Using the conservation of momentum, we can write the equation:
(5.0 kg × v1') + (4.5 kg × 0.0 m/s) = 55.0 kg·m/s
Simplifying the equation, we have:
5.0 kg × v1' = 55.0 kg·m/s
Dividing both sides by 5.0 kg:
v1' = 55.0 kg·m/s / 5.0 kg
v1' = 11.0 m/s
Therefore, the velocity of block 1 after the collision is -11.0 m/s. Since the positive direction was defined as motion to the right, the negative sign indicates that block 1 is now moving to the left with a velocity of 11.0 m/s.
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Write the electric field of a dipole in vector notation. Using the result of Problem 3, find the potential energy of a dipole of moment d in the field of another dipole of moment d'. (Take d' at the origin and d at position r.) Find the forces and couples acting between the dipoles if they are placed on the z-axis and (a) both are pointing in the z- direction, (b) both are pointing in the x-direction, (c) d is in the z- direction, and d' in the x-direction, and (d) d is in the x-direction and d' in the y-direction.
The electric field of a dipole in vector notation is given by E = (k * p) / r^3, where E is the electric field, k is the electrostatic constant, p is the dipole moment, and r is the distance from the dipole.
To find the potential energy of a dipole of moment d in the field of another dipole of moment d', we can use the formula U = -p * E, where U is the potential energy, p is the dipole moment, and E is the electric field. To find the forces and couples acting between the dipoles in different orientations, we need to consider the interaction between the electric fields and the dipole moments.
(a) When both dipoles are pointing in the z-direction, the forces between them will be attractive, causing the dipoles to come together along the z-axis.
(b) When both dipoles are pointing in the x-direction, there will be no forces or couples acting between them since the electric field and the dipole moment are perpendicular.
(c) When d is in the z-direction and d' is in the x-direction, the forces between them will be attractive along the z-axis, causing the dipoles to align in that direction.
(d) When d is in the x-direction and d' is in the y-direction, there will be no forces or couples acting between them since the electric field and the dipole moment is perpendicular.
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Volcanoes on Io. Io, a satellite of Jupiter, is the most volcanically active moon or planet in the solar system. It has volcanoes that send plumes of matter over 500 km high (see Figure 7.45). Due to the satellite’s small mass, the acceleration due to gravity on Io is only 1.81 m>s 2, and Io has no appreciable atmosphere. Assume that there is no variation in gravity over the distance traveled. (a) What must be the speed of material just as it leaves the volcano to reach an altitude of 500 km? (b) If the gravitational potential energy is zero at the surface, what is the potential energy for a 25 kg fragment at its maximum height on Io? How much would this gravitational potential energy be if it were at the same height above earth?
(a) Therefore, the speed of material just as it leaves the volcano to reach an altitude of 500 km is 2000 m/s. (b) Thus, the gravitational potential energy of the volcanic fragment when it is at the same height above Earth would be 12,262,500 J.
(a)The potential energy gained by the volcanic material in the process of rising to 500 km altitude is provided by the decrease in gravitational potential energy.
The formula for potential energy is given by:-PE = mgh Where, m = mass of the volcanic matter g = acceleration due to gravity h = height of the volcanic matter above the surface of the satellite
Here, m = mass of volcanic matter (unknown)g = acceleration due to gravity on Io = 1.81 m/s²h = height of volcanic matter above the surface of the satellite = 500 km = 500,000 m
The potential energy is equal to the work done by gravity, so the gain in potential energy equals the loss in kinetic energy.
The volcanic material loses all its initial kinetic energy at a height of 500 km above Io
So, KE = 1/2 mv²Where,v = velocity of volcanic material. We can equate the potential energy gained by the volcanic material with the initial kinetic energy of the volcanic material.
That is,mgh = 1/2 mv²hence,v = √(2gh) = √(2 × 1.81 m/s² × 500,000 m) = 2000 m/s
Therefore, the speed of material just as it leaves the volcano to reach an altitude of 500 km is 2000 m/s.
(b)The formula for potential energy is given by:-PE = mgh Where,m = mass of the volcanic fragment g = acceleration due to gravityh = height of the volcanic fragment above the surface of the satellite
Here, m = 25 kgg = acceleration due to gravity on Io = 1.81 m/s²h = height of the volcanic fragment above the surface of the satellite = 500 km = 500,000 mPE = mgh = 25 × 1.81 m/s² × 500,000 m = 22,625,000 J
When the volcanic fragment is at the same height above the Earth, its gravitational potential energy would be given by the same formula, except the acceleration due to gravity would be that at Earth's surface, which is 9.81 m/s².
Therefore,-PE = mgh = 25 × 9.81 m/s² × 500,000 m = 12,262,500 J
Thus, the gravitational potential energy of the volcanic fragment when it is at the same height above Earth would be 12,262,500 J.
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A single conducting loop of wire has an area of 7.4x10-2 m² and a resistance of 120 Perpendicular to the plane of the loop is a magnetic field of strength 0.55 T. Part A At what rate (in T/s) must this field change if the induced current in the loop is to be 0.40 A
The rate of change of the magnetic field is 48 T/s in the direction opposite to the magnetic field. Answer: -48 T/s
A single conducting loop of wire has an area of 7.4 x 10-2 m² and a resistance of 120 Ω. Perpendicular to the plane of the loop is a magnetic field of strength 0.55 T. To find the rate of change of magnetic field, we can use Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit. The magnetic flux through the loop is given by:ΦB = B A cos θWhere B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the plane of the loop.
Since the magnetic field is perpendicular to the plane of the loop, θ = 90°. Therefore,ΦB = B A cos 90° = 0.55 x 7.4 x 10-2 = 0.0407 T m²The induced emf in the loop is given by:emf = - N dΦB / dtwhere N is the number of turns in the loop and dΦB / dt is the rate of change of the magnetic flux through the loop.The negative sign in the equation is due to Lenz's law, which states that the direction of the induced emf is such that it opposes the change in magnetic flux that produces it.Since there is only one turn in the loop, N = 1.
Therefore,emf = - dΦB / dtIf the induced current in the loop is to be 0.40 A, then we have:emf = IRwhere I is the induced current and R is the resistance of the loop.Rearranging this equation, we get:dΦB / dt = - (IR)Substituting the given values, we get:dΦB / dt = - (0.40) x (120) = - 48 T/sSince the magnetic field is changing in time, we have to include the sign of the rate of change of the magnetic flux. The negative sign indicates that the magnetic field is decreasing in strength with time. Therefore, the rate of change of the magnetic field is 48 T/s in the direction opposite to the magnetic field. Answer: -48 T/s
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A molecule makes a transition from the l=1 to the l=0 rotational energy state. When the wavelength of the emitted photon is 1.0×10 −3
m, find the moment of inertia of the molecule in the unit of kg m 2
.
The moment of inertia of the molecule in the unit of kg m2 is 1.6 × 10-46.
The energy difference between rotational energy states is given by
ΔE = h² / 8π²I [(l + 1)² - l²] = h² / 8π²I (2l + 1)
For l = 1 and l = 0,ΔE = 3h² / 32π²I = hc/λ
Where h is the Planck constant, c is the speed of light and λ is the wavelength of the emitted photon.
I = h / 8π²c
ΔEλ = h / 8π²c (3h² / 32π²I )λ = 3h / 256π³cI = 3h / 256π³cλI = (3 × 6.626 × 10-34)/(256 × (3.1416)³ × (3 × 108))(1.0×10 −3 )I = 1.6 × 10-46 kg m2
Hence, the moment of inertia of the molecule in the unit of kg m2 is 1.6 × 10-46.
Answer: 1.6 × 10-46
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625 C passes through a flashlight in 0.460 h. What is the
average current?
625 C passes through a flashlight in 0.460 h. the average current passing through the flashlight is approximately 0.377 A.
To calculate the average current, we need to use the formula:
Average Current (I) = Total Charge (Q) / Time (t)
In this case, we are given that a total charge of 625 C passes through the flashlight. The time is given as 0.460 hours.
First, we need to convert the time from hours to seconds since the unit of current is in amperes (A), which is defined as coulombs per second.
0.460 hours is equal to 0.460 x 60 x 60 = 1656 seconds.
Now we can calculate the average current:
I = 625 C / 1656 s
I ≈ 0.377 A
Therefore, the average current passing through the flashlight is approximately 0.377 A.
Average current is a measure of the rate at which charge flows through a circuit over a given time. In this case, the average current tells us how much charge, in coulombs, passes through the flashlight per second. It is an important parameter to consider when analyzing the behavior and performance of electrical devices.
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A block slides down a ramp with an incline of 45 degrees, a distance of 50 cm along the ramp at constant velocity. If the block has a mass of 1.5 kg, how much thermal energy was produced by friction during this descent? Use g= 10 m/s2
The work done by friction represents the thermal energy produced during the descent of the block. Therefore, the thermal energy produced by friction is 1.591 Joules.
To determine the thermal energy produced by friction during the descent of the block, we need to calculate the work done by friction and convert it into thermal energy.
The work done by friction can be calculated using the equation:
Work = Force of friction x Distance
The force of friction can be found using the equation:
Force of friction = Normal force x Coefficient of friction
The normal force acting on the block can be determined using the equation:
Normal force = mass x gravitational acceleration x cosine(angle of incline)
In this case, the angle of incline is 45 degrees, and the gravitational acceleration (g) is given as 10 m/s^2.
First, let's calculate the normal force:
Normal force = 1.5 kg x 10 m/s^2 x cos(45 degrees)
Normal force = 1.5 kg x 10 m/s^2 x 0.707
Normal force = 10.606 N
Next, we can calculate the force of friction using the coefficient of friction. Let's assume a coefficient of friction of 0.3 (this value depends on the surfaces in contact):
Force of friction = Normal force x Coefficient of friction
Force of friction = 10.606 N x 0.3
Force of friction = 3.182 N
Now, we can calculate the work done by friction:
Work = Force of friction x Distance
Work = 3.182 N x 0.5 m (converting 50 cm to 0.5 m)
Work = 1.591 J
The work done by friction represents the thermal energy produced during the descent of the block. Therefore, the thermal energy produced by friction is 1.591 Joules.
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A uniform rod, supported and pivoted at its midpoint, but initially at rest, has a mass of 73 g and a length 2 cm. A piece of clay with mass 28 g and velocity 2.3 m/s hits the very top of the rod, gets stuck and causes the clayrod system to spin about the pivot point O at the center of the rod in a horizontal plane. Viewed from above the scheme is With respect to the pivot point O, what is the magnitude of the initial angular mo- mentum L i
of the clay-rod system? After the collisions the clay-rod system has an angular velocity ω about the pivot. Answer in units of kg⋅m 2
/s. 007 (part 2 of 3 ) 10.0 points With respect to the pivot point O, what is the final moment of inertia I f
of the clay-rod system? Answer in units of kg⋅m 2
. 008 (part 3 of 3) 10.0 points What is the final angular speed ω f
of the clay-rod system? Answer in units of rad/s.
1. The magnitude of the initial angular momentum (Li) of the clay-rod system about the pivot point O can be calculated by considering the individual angular momenta of the clay and the rod., 2. The final moment of inertia (If) of the clay-rod system after the collision can be determined by adding the moments of inertia of the clay and the rod. 3. The final angular speed (ωf) of the clay-rod system can be calculated using the conservation of angular momentum.
1. The initial angular momentum (Li) of the clay-rod system about the pivot point O can be calculated as the sum of the angular momentum of the clay and the angular momentum of the rod. The angular momentum of an object is given by the product of its moment of inertia and angular velocity.
2. The final moment of inertia (If) of the clay-rod system is obtained by adding the moments of inertia of the clay and the rod. The moment of inertia depends on the mass and distribution of mass of the object.
3. Using the conservation of angular momentum, we can equate the initial angular momentum (Li) to the final angular momentum (Lf), and solve for the final angular speed (ωf). The conservation of angular momentum states that the total angular momentum of a system remains constant if no external torque acts on it.
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An atom with 276 nucleons, of which 121 are protons, has a mass of 276.1450 u. What is the binding energy per nucleon of the nucleons in its nucleus? The mass of a hydrogen atom is 1.007825 u and the mass of a neutron is 1.008665 u. Number ____________ Units ____________
The binding energy per nucleon of the nucleons in an atom with 276 nucleons, of which 121 are protons, has a mass of 276.1450 u is 7.21 MeV/nucleon.
We are given the following data: 276 nucleons 121 protons. The total number of neutrons in the atom can be determined by subtracting the number of protons from the total number of nucleons.276 - 121 = 155Thus, there are 155 neutrons in the atom. The mass of the nucleus can be computed as follows: Mass of nucleus = (121 * 1.007825) + (155 * 1.008665)= 122.357525 + 156.395075= 278.7526 u. The mass defect of the nucleus can be calculated using the following equation: mass defect = (number of protons * mass of proton) + (number of neutrons * mass of neutron) - mass of nucleus mass defect = (121 * 1.007825) + (155 * 1.008665) - 276.1450mass defect = 1.290725 u.
The binding energy of the nucleus can now be calculated using the following equation: binding energy = mass defect * c²where c is the speed of light (299792458 m/s)binding energy = 1.290725 * (299792458)²= 1.1607 × 10²¹ J/nucleon = 7.21 MeV/nucleon Number = 7.21 Units = MeV/nucleon.
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Using the loop rule and deriving the differential equation for an LC circuit find the current (sign included) through the inductor at the instant t = 1.2 s if L = 2.7 H, C = 3.3 F. The initial charge at the capacitor is Qo = 4.30 and the initial current through the inductor is lo=0. Number Units
The current (sign included) through the inductor at the instant t is -0.089 A (negative sign implies that the current direction is opposite to the assumed direction).
How to determine current?The loop rule in an LC circuit gives us the equation Q/C + L×dI/dt = 0. Using the fact that I = dQ/dt, differentiate both sides to obtain:
d²Q/dt² + 1/(LC)Q = 0
This is a simple harmonic oscillator equation. The general solution is:
Q(t) = A cos(wt + φ)
where w = √(1/LC) is the angular frequency, A is the amplitude, and φ is the phase.
Given that Q(0) = Qo = 4.30, so:
A cos(φ) = Qo
Also given that I(0) = dQ/dt(0) = Io = 0. So differentiating Q(t) and setting t = 0 gives:
-Aw sin(φ) = Io
From these two equations solve for A and φ. The second equation tells us that sin(φ) = 0, so φ is 0 or pi. Since cos(0) = 1 and cos(pi) = -1, and A must be positive (since it's an amplitude), we choose φ = 0. This gives:
A = Qo
So the solution is:
Q(t) = Qo cos(wt)
and hence
I(t) = dQ/dt = -w Qo sin(wt)
Substitute w = √(1/LC), Qo = 4.30, and t = 1.2s:
I(1.2) = - √(1/(2.73.3)) × 4.3 × sin( √(1/(2.73.3)) × 1.2)
Doing the arithmetic, this gives:
I(1.2) = -0.089 A
The negative sign implies that the current direction is opposite to the assumed direction.
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