The wavelength of the EM wave is 0.3 meters (or 30 centimeters).
The frequency of an electromagnetic wave is 10⁶ Hz. Find the wavelength of this EM wave.The velocity of light in a vacuum is 3 x 10⁸ m/s.
The formula for the wavelength is given by;
Wavelength (λ) = Speed of light (c) / Frequency (f)
λ = c / f= 3 x 10⁸ m/s / 10⁶ Hz = 300 m/s ÷ 10⁶ Hz= 0.3 meters or 30 centimeters
Therefore, the wavelength of the EM wave is 0.3 meters (or 30 centimeters).
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An object is placed 10cm in front of a concave mirror whose radius of curvature is 10cm calculate the position ,nature and magnification of the image produced
Answer:
The focal length, f = − 15 2 c m = − 7.5 c m The object distance, u = -10 cm Now from the mirror equation 1 v + 1 u = 1 f 1 v + 1 − 10 = 1 − 7.5 v = 10 × 7.5 − 2.5 = − 30 c m The image is 30 cm from the mirror on the same side as the object.
A 1900 kg car accelerates from 12 m/s to 20 m/s in 9 s. The net force acting on the car is:
The 1900 kg car accelerates from 12 m/s to 20 m/s in 9 seconds. We need to determine the net force acting on the car is 1691 N.
To find the net force acting on the car, we can use Newton's second law of motion, which states that the net force on an object is equal to the object's mass multiplied by its acceleration
[tex](F_net = m * a)[/tex]
First, we calculate the acceleration of the car using the equation
[tex]a = (v_f - v_i) / t[/tex]
where v_f is the final velocity, v_i is the initial velocity, and t is the time taken. Plugging in the given values, we have
[tex]a = (20 m/s - 12 m/s) / 9 s = 0.89 m/s^2.[/tex]
Next, we can calculate the net force by multiplying the mass of the car by its acceleration:
[tex]F_net = 1900 kg * 0.89 m/s^2 = 1691 N.[/tex]
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The magnetic field is 1.50 uT at a distance 42.6 cm away from a long, straight wire. At what distance is it 0.150 uT? 4.26×10 2
cm Previous Tries the middle of the straight cord, in the plane of the two wires. Tries 2/10 Previous Tries
The distance from the wire is 426 cm is the answer.
Given data: The magnetic field, [tex]B = 1.50 uT[/tex]
The distance from the long, straight wire, [tex]r1 = 42.6 cm.[/tex]
The magnetic field,[tex]B' = 0.150 uT[/tex]
To find: the distance from the wire, r2
Solution: We can use the Biot-Savart law to find the magnetic field at a distance r from an infinitely long straight wire carrying current I: [tex]B = μ0I / 2πr[/tex] where [tex]μ0 = 4π[/tex]× [tex]10^-7[/tex] Tm/A is the permeability of free space.
Now we can write this as: [tex]r = μ0I / 2πB[/tex] .....(1)
At [tex]r1, B = 1.50 uT[/tex] and at[tex]r2, B' = 0.150 uT[/tex]
Therefore, from equation (1):[tex]r2 = μ0I / 2πB'[/tex].....(2)
Let us assume the current in the wire is I. Since I is constant, we can write [tex]r2/r1 = B / B'.[/tex]....(3)
Substituting the values:[tex]r2 / 42.6 = 1.50 / 0.150[/tex]
Solving for [tex]r2:r2 = (42.6 × 1.50) / 0.150 = 426 cm[/tex]
Therefore, the magnetic field is 0.150 uT at a distance of 426 cm from the wire.
Thus, the distance from the wire is 426 cm.
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For the systems whose closed loop transfer functions are given below, determine whether the system is stable, marginally stable or unstable. -5s +3 2s-1 a) T₁(s)=- 2s +1 (s+1)(s²-3s+2)' ; b) T₂ (s)=- (5+1)(s² + s +1)* ) ₂ (s) = (s-2)(s² +s+1)' 2s+1 d) T₁ (s)=- ; e) T,(s) = (s+1)(s² +1)' f)T(s)=- s+5 (s+3)(x²+4)² s-1 s(s² + s +1)
We aim to prove that the functions f(x) and x*f(x) are linearly independent for any non-constant function f(x). Linear independence means that no non-trivial linear combination of the two functions can result in the zero function.
By assuming the existence of constants a and b, we will demonstrate that the only solution to the equation a*f(x) + b*(x*f(x)) = 0 is a = b = 0. To begin, let's consider the linear combination a*f(x) + b*(x*f(x)) = 0, where a and b are constants. We want to show that the only solution to this equation is a = b = 0.
Expanding the expression, we have a*f(x) + b*(x*f(x)) = (a + b*x)*f(x) = 0. Since f(x) is a non-constant function, there exists at least one value of x (let's call it x0) for which f(x0) ≠ 0.Plugging in x = x0, we obtain (a + b*x0)*f(x0) = 0. Since f(x0) ≠ 0, we can divide both sides of the equation by f(x0), resulting in a + b*x0 = 0.
Now, notice that this linear equation holds for all x, not just x0. Therefore, a + b*x = 0 is true for all x. Since the equation is linear, it must hold for at least two distinct values of x. Let's consider x1 ≠ x0. Plugging in x = x1, we have a + b*x1 = 0.Subtracting the equation a + b*x0 = 0 from the equation a + b*x1 = 0, we get b*(x1 - x0) = 0. Since x1 ≠ x0, we have (x1 - x0) ≠ 0. Therefore, b must be equal to 0.
With b = 0, we can substitute it back into the equation a + b*x0 = 0, giving us a + 0*x0 = 0. This simplifies to a = 0. Hence, we have shown that the only solution to the equation a*f(x) + b*(x*f(x)) = 0 is a = b = 0. Therefore, the functions f(x) and x*f(x) are linearly independent for any non-constant function f(x).In conclusion, the functions f(x) and x*f(x) are linearly independent because their only possible linear combination resulting in the zero function is when both the coefficients a and b are equal to zero.
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△50% Part (a) What is the oscillation frequency of your circuit, in hertz? A 50% Part (b) If the maximum potential difference between the plates of the capacitor is 55 V, what is the maximum current in the circuit, in amperes? I max
=
Therefore, we cannot determine the values for parts (a) and (b) of the question. Unfortunately, we cannot determine the values for parts (a) and (b) of the question.
For a parallel-plate capacitor, the capacitance, C is given byC=ϵ0A/dwhere ϵ0 is the permittivity of free space, A is the area of each plate, and d is the distance between the plates. The period of oscillation is given byT=2π√LCwhere L is the inductance of the inductor in the circuit. Since the circuit oscillates at 50% of its maximum value, the peak current, I_max can be determined usingOhm's law, I=V/R. The current, I at any given moment in time can be found usingI=I_maxsin(ωt), where ω is the angular frequency, which is given byω=2π/T. Part (a)The oscillation frequency of the circuit, in hertz, is given byf=1/T=1/2π√LC. Since we are not given any values for the inductance or capacitance, we cannot determine the frequency of oscillation. Part (b)The maximum current, I_max, is given byI_max=V/R, where V is the maximum potential difference between the plates of the capacitor and R is the resistance of the circuit. We are not given any information about the resistance of the circuit, so we cannot determine the maximum current in amperes. Therefore, we cannot determine the values for parts (a) and (b) of the question. Answer: Unfortunately, we cannot determine the values for parts (a) and (b) of the question.
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Write True if the statement is correct, but if it's False, change the underlined word
or group of words to make the whole statement true. Write your answer on the space
provided before the number.
1. Heat engine is a device that converts thermal energy into mecha
work.
2. Doing mechanical work on the system will decrease its internal energy.
3. Internal energy is proportional to the change in temperature.
4. Only heat contributes to the total internal energy of the system.
5. Internal energy stored in the body is in the form of fats.
6. Heat can be completely transformed into work.
7. If the amount of work done (W) is the same as the amount of energy
transferred in by heat (Q), the net change in internal energy is 1.
8. The temperature of the system increases when work is done on the
system.
1. True. A heat engine is a device that converts thermal energy into mecha work.
2. False. Doing mechanical work on the system will increase or decrease its internal energy.
3. False. Internal energy is not directly proportional to the change in temperature.
4. False. Both heat and work can contribute to the total internal energy of the system.
5. False. Internal energy stored in the body is in the form of various energy sources.
6. False. Heat cannot be completely transformed into work without any losses.
7. False. If the amount of work done (W) is the same as the amount of energy transferred in by heat (Q), the net change in internal energy is zero.
8. False. The temperature of the system may increase or decrease when work is done on the system.
1. A heat engine is a device that converts thermal energy into mecha work.
2. False. Doing mechanical work on the system can either increase or decrease its internal energy, depending on the specific circumstances.
3. False. Internal energy is not directly proportional to the change in temperature. It depends on various factors such as pressure, volume, and the type of substance. The change in internal energy can be influenced by multiple factors, not just temperature.
4. False. Both heat and work can contribute to the total internal energy of the system. Internal energy is the sum of the system's kinetic energy and potential energy, which can be affected by both heat and work interactions.
5. False. Internal energy stored in the body is not solely in the form of fats. It includes various forms of energy, including chemical energy from nutrients, thermal energy, and other forms.
6. False. Heat cannot be completely transformed into work without any losses according to the laws of thermodynamics. There will always be some inefficiencies and losses in the conversion process.
7. False. If the amount of work done (W) is the same as the amount of energy transferred in by heat (Q), the net change in internal energy is zero according to the first law of thermodynamics, not 1.
8. False. The temperature of the system can increase or decrease when work is done on the system, depending on various factors such as the type of work done, the properties of the system, and the specific conditions.
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A cauterizer, used to stop bleeding in surgery, puts out 1.75 mA at 16.0kV. (a) What is its power output (in W)? W (b) What is the resistance (in MΩ ) of the path? \& MΩ
a) The power output of the cauterizer is 28 W.b) The resistance of the path is 9.14 MΩ.
(a) To find the power output of the cauterizer, we can use the formula:Power (P) = Voltage (V) x Current (I)orP = VIWe are given the voltage and current, so we can substitute the values:P = (16.0 kV)(1.75 mA) = 28 WTherefore, the power output of the cauterizer is 28 W.
(b) To find the resistance of the path, we can use Ohm's law:V = IRRearranging the formula, we get:I = V/RSubstituting the values we have:1.75 mA = 16.0 kV / RConverting the units of current to amperes:1.75 x 10^-3 A = 16,000 V / RDividing both sides by 1.75 x 10^-3 A:R = (16,000 V) / (1.75 x 10^-3 A)R = 9,142,857 Ω = 9.14 MΩTherefore, the resistance of the path is 9.14 MΩ.
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A clock has a 10.0-g mass object bouncing on a spring that has a force constant of 0.9 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? Umax m/s How many joules of kinetic energy does the object have at its maximum velocity? KEmax x 10-4 -
A clock has a 10.0-g mass object bouncing on a spring that has a force constant of 0.9 N/m. the object has approximately 1.08 x 10^(-3) J of kinetic energy at its maximum velocity.
To find the maximum velocity of the object bouncing on the spring, we can use the principle of conservation of mechanical energy.
The maximum potential energy of the object can be calculated when it reaches its maximum displacement from the equilibrium position. Since the object bounces 3.00 cm above and below the equilibrium position, the total displacement is 2 * 3.00 cm = 6.00 cm = 0.06 m.
The maximum potential energy can be calculated using the equation:
PE_max = 0.5 * k * x^2,
where k is the force constant of the spring and x is the maximum displacement.
Substituting the given values:
PE_max = 0.5 * 0.9 N/m * (0.06 m)^2
= 0.00108 J
According to the conservation of mechanical energy, this potential energy is converted into kinetic energy when the object reaches its maximum velocity.
Therefore, the kinetic energy at maximum velocity is equal to the potential energy:
KE_max = 0.00108 J
In scientific notation, KE_max ≈ 1.08 x 10^(-3) J.
Therefore, the object has approximately 1.08 x 10^(-3) J of kinetic energy at its maximum velocity.
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A sailor uses an ultrasonic crack detector to find flaws in the rubber gasket ( S.G = 2.4, Y = 2.5 GPa) sealing water tight compartments. The crack detector produces 21.06 KHz pulses.
a) Calculate the speed of sound in the gasket in m/s
b) Calculate the wavelength
c) A crack is thought to be at a depth of 1.874 cm. Calculate the expected interval time for the pulse to make a round rip in μs.
The expected interval time for the pulse to make a round trip in the gasket is approximately 22.7 μs.
To calculate the speed of sound in the gasket, we can use the formula:
Speed of sound = Frequency × Wavelength
a) Calculate the speed of sound in the gasket in m/s:
Given:
Frequency = 21.06 KHz = 21.06 × 10^3 Hz
To calculate the speed of sound, we need the wavelength. Since the wavelength is not given directly, we can use the following formula to find it:
Wavelength = Speed of sound / Frequency
We know that the speed of sound in a material is given by:
Speed of sound = √(Young's modulus / Density)
Given:
Young's modulus (Y) = 2.5 GPa = 2.5 × 10^9 Pa
Density (ρ) = Specific gravity (SG) × Density of water
Density of water = 1000 kg/m^3 (approximate value)
Specific gravity (SG) = 2.4
Density (ρ) = 2.4 × 1000 kg/m^3 = 2400 kg/m^3
Now, we can substitute these values to calculate the speed of sound:
Speed of sound = √(2.5 × 10^9 Pa / 2400 kg/m^3)
= √(2.5 × 10^9 / 2400) m/s
≈ 1650.82 m/s
b) Calculate the wavelength:
Wavelength = Speed of sound / Frequency
= 1650.82 m/s / (21.06 × 10^3 Hz)
≈ 78.34 × 10^-6 m
≈ 78.34 μm
c) Calculate the expected interval time for the pulse to make a round trip in μs:
Given:
Depth of crack = 1.874 cm = 1.874 × 10^-2 m
The time taken for a round trip can be calculated as:
Round trip time = 2 × Depth of crack / Speed of sound
Round trip time = 2 × (1.874 × 10^-2 m) / 1650.82 m/s
≈ 2.27 × 10^-5 s
≈ 22.7 μs
Therefore, the expected interval time for the pulse to make a round trip in the gasket is approximately 22.7 μs.
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What is the magnetic field strength created at its center in T ?
The magnetic field strength created at the center of a circular loop carrying a current of 30.0 A and consisting of 250 turns with a radius of 10.0 cm is approximately 3.8 × 10^(-3) T (tesla).
The magnetic field strength at the center of a circular loop carrying current can be calculated using the formula: B = (μ₀ * I * N) / (2 * R), where B is the magnetic field strength, μ₀ is the permeability of free space (approximately 4π × 10^(-7) T·m/A), I is the current, N is the number of turns in the loop, and R is the radius of the loop.
Substituting the given values, we have:
B = (4π × 10^(-7) T·m/A * 30.0 A * 250) / (2 * 0.10 m)
B ≈ 3.8 × 10^(-3) T
Therefore, the magnetic field strength created at the center of the circular loop is approximately 3.8 × 10^(-3) T (tesla).
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The complete question is:
Inside a motor, 30.0 A passes through a 250 -turn circular loop that is 10.0 cm in radius. What is the magnetic field strength created at its center?
A 0.480-kg pendulum bob passes through the lowest part of its path at a speed of 7.46 m/s. (a) What he the magnitude of the tension in the pendulum cable at this point if the pendulum is 79.0 cm lang? N (b) When the pendolum feaches its highest point, what angle does the cable make with the vertical? (Enter your answer to at least ane decimat phace.) (c) What is the magnitude of the tertion in the pendulum cable when the pendulum reaches its highest point? P
(a) Mv²/2 = mgh where v = 7.46 m/s, m = 0.480 kg, g = 9.81 m/s²,h = 0.79 m. (b) Thus, sinθ = opposite/hypotenuse = 0.79/h , Hypotenuse = length of the pendulum = 0.79 m. (c) Thus, the magnitude of the tension in the pendulum cable is 4.71 N
a) Magnitude of tension in the pendulum cable: 56.58 N When the pendulum bob is at its lowest point, all its energy will be in the form of kinetic energy.
Thus, it can be stated that KE + PE = constant.
Here, PE is zero as there is no height, and thus the total energy of the system is equal to the kinetic energy of the pendulum bob.Mv²/2 = mgh wherev = 7.46 m/s, m = 0.480 kg,g = 9.81 m/s²,h = 0.79 m
By substituting these values in the above formula, we get: Tension in the pendulum cable is equal to weight component in the direction of the cable, which is given by: mg cosθ
Here,θ is the angle the cable makes with the vertical.
b) The angle that the cable makes with the vertical is: 64.67°When the pendulum bob is at its highest point, all its energy will be in the form of potential energy.
Thus, it can be stated that KE + PE = constant.
Here, KE is zero as there is no motion, and thus the total energy of the system is equal to the potential energy of the pendulum bob. mgh = mgh wherev = 0 m/s,m = 0.480 kg, g = 9.81 m/s²,h = 0.79 m
Thus, sinθ = opposite/hypotenuse = 0.79/h , Hypotenuse = length of the pendulum = 0.79 m
c) Magnitude of tension in the pendulum cable: 4.59 N
At the highest point, the tension in the cable is equal to the weight of the bob, which is given by:mg = 0.480 × 9.81 = 4.7068 N
Thus, the magnitude of the tension in the pendulum cable is 4.71 N (rounded off to two decimal places).
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Since the investigative question has two variables, you need to focus on each one separately. Thinking only about the first part of the question, mass, what might be a hypothesis that would illustrate the relationship between mass and kinetic energy? Use the format of "if…then…because…” when writing your hypothesis.
In order to form a hypothesis that would illustrate the relationship between mass and kinetic energy, we first need to understand what kinetic energy and mass are and how they are related. Kinetic energy is the energy that an object possesses due to its motion, and is given by the formula KE = 0.5mv², where m is the mass of the object and v is its velocity. Mass, on the other hand, is a measure of the amount of matter in an object.
The relationship between mass and kinetic energy is direct, meaning that as mass increases, so does kinetic energy, provided that velocity remains constant. Similarly, if velocity increases, then kinetic energy will increase as well, provided that mass remains constant.
The hypothesis that illustrates this relationship can be stated as follows:If the mass of an object is increased, then the kinetic energy of the object will also increase, because kinetic energy is directly proportional to mass, assuming velocity remains constant.In other words, if the mass of an object is doubled, then its kinetic energy will also double, assuming that its velocity remains constant. This hypothesis can be tested through experiments that involve measuring the kinetic energy of objects with different masses, but with the same velocity.
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If the mass of an object increases, then its kinetic energy will increase proportionally because mass and kinetic energy have a linear relationship when graphed.
What does it cost to cook a chicken for 1 hour in an oven that operates at 20 Ampere and 220 Volt if the electric company charge 40 fils per kWh A. 264 Fils B. 528 Fils C. 352 Fils D. 176 Fils
The cost to cook a chicken for 1 hour in the given oven is 176 fils. When charged electrons (current) are forced through a conducting loop by the pressure of an electrical circuit's power source, they may perform tasks like lighting a lamp. In a nutshell, voltage is equal to pressure and is expressed in volts (V).
To calculate the cost of cooking a chicken for 1 hour in the given oven, we need to determine the power consumption of the oven.
Power (P) can be calculated using the formula:
P = V * I
where V is the voltage (220 V) and I is the current (20 A).
P = 220 V * 20 A = 4400 W
Now, we convert the power from watts to kilowatts:
P_kW = P / 1000 = 4400 W / 1000 = 4.4 kW
To calculate the cost, we multiply the power consumption by the time (1 hour) and the cost per kilowatt-hour:
Cost = P_kW * time * cost per kWh
Cost = 4.4 kW * 1 hour * 40 fils/kWh
Cost = 4.4 * 1 * 40 = 176 fils
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A buzzer attached cart produces the sound of 620 Hz and is placed on a moving platform. Ali and Bertha are positioned at opposite ends of the cart track. The platform moves toward Ali while away from Bertha. Ali and Bertha hear the sound with frequencies f₁ and f2, respectively. Choose the correct statement. A. (f₁f2) > 620 Hz B. fi > 620 Hz > f₂ C. f2> 620 Hz > f₁
Ali hears a higher frequency than the emitted frequency (620 Hz) and Bertha hears a lower frequency than the emitted frequency, the correct statement is C. f₂ > 620 Hz > f₁.
When a sound source is moving towards an observer, the frequency of the sound heard by the observer is higher than the actual frequency emitted by the source. This phenomenon is known as the Doppler effect. Conversely, when a sound source is moving away from an observer, the frequency of the sound heard is lower than the actual frequency emitted.
In this scenario, as the buzzer attached to the cart is placed on a moving platform and is approaching Ali while moving away from Bertha, Ali will hear a higher frequency f₁ compared to the emitted frequency of 620 Hz. On the other hand, Bertha will hear a lower frequency f₂ compared to the emitted frequency of 620 Hz.
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In a photoelectric effect experiment, if the frequency of the photons are increased while the intensity of the photons are held the same. the work function increases. the maximum kinetic energy of the photoelectrons increases. the maximum current increases. the stopping potential decreases.
The correct option is b. Increasing the frequency of photons in a photoelectric effect experiment while keeping the intensity constant will result in an increase in the maximum kinetic energy of the photoelectrons.
The photoelectric effect refers to the emission of electrons from a material when it is exposed to light. The energy of the emitted electrons is determined by the frequency of the photons that strike the material.
According to the equation E = hf, where E is the energy of a photon, h is Planck's constant, and f is the frequency of the photon, increasing the frequency of photons will lead to an increase in the energy of the individual photons. Therefore, when the frequency is increased while the intensity (number of photons per second) remains constant, the average energy of the photons increases.
The maximum kinetic energy of the photoelectrons depends on the energy of the incident photons and the work function of the material, which is the minimum energy required for an electron to be emitted. As the frequency of the photons increases, the energy of the photons increases, resulting in a higher maximum kinetic energy for the emitted electrons. Therefore, the correct option is b) the maximum kinetic energy of the photoelectrons increases.
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The complete question is:
In a photoelectric effect experiment, if the frequency of the photons is increased while the intensity of the photons is held the same. Choose the option which is best suitable
a)the work function increases.
b)the maximum kinetic energy of the photoelectrons increases.
c)the maximum current increases.
d)the stopping potential decreases.
It is desired to sample, by means of an ADC, any signal for which the following data is known: The maximum power of the signal reaches 800 mW The minimum power is 0.1 mW. Its maximum frequency reaches 10 kHz.
Determine:
a) The dynamic range (DR) of the signal.
b) The minimum number of bits of resolution (of the ADC) required to avoid distortion and that meets
with the SNR.
c) The conversion time required to satisfy the maximum frequency of the signal
a) The dynamic range (DR) of the signal is approximately 33.98 dB.
b) The minimum number of bits of resolution required for the ADC is 11 bits.
c) The conversion time required to satisfy the maximum frequency of the signal is 0.1 milliseconds.
a) The dynamic range (DR) of a signal is the ratio between the maximum and minimum power levels, expressed in decibels (dB). In this case, the dynamic range can be calculated using the formula DR = 10 * log10(maximum power/minimum power), which results in DR ≈ 33.98 dB.
b) The minimum number of bits of resolution required for the ADC can be determined based on the desired signal-to-noise ratio (SNR). The formula to calculate the required number of bits is N = ceil(log2(4 * SNR)), where SNR is the desired signal-to-noise ratio. Assuming a desired SNR of 6 dB, the minimum number of bits required would be N ≈ 11.
c) The conversion time required to satisfy the maximum frequency of the signal can be determined using the Nyquist-Shannon sampling theorem, which states that the sampling rate should be at least twice the maximum frequency. Therefore, the conversion time can be calculated as 1 / (2 * maximum frequency), resulting in a conversion time of approximately 0.1 milliseconds.
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The position vector of a particle of mass 2.20 kg as a function of time is given by ř = (6.00 i + 5.40 tſ), whereř is in meters and t is in seconds. Determine the angular momentum of the particle about the origin as a function of time. k) kg · m²/s
The angular momentum of the particle about the origin as a function of time is L = (32.40)k kg · m²/s. The angular momentum does not depend on time and remains constant throughout the motion.
The angular momentum of a particle about the origin is given by L = m(ř × v), where m is the mass of the particle, ř is the position vector, and v is the velocity vector. To calculate the angular momentum as a function of time, we need to find the time derivative of the position vector and the velocity vector.
Given that ř = (6.00 i + 5.40 t), the velocity vector v is the derivative of ř with respect to time: v = dř/dt = (0 + 5.40) i = 5.40 i m/s.
Now we can calculate the cross product of ř and v. The cross product of two vectors in three dimensions is given by the formula (a × b) = (a_yb_z - a_zb_y)i + (a_zb_x - a_xb_z)j + (a_xb_y - a_yb_x)k. In this case, since both vectors ř and v have only i-components, the cross product simplifies to L = m(0 - 0)i + (0 - 0)j + (6.00 * 5.40 - 0)k = (0)i + (0)j + (32.40)k.
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An object with a mass of 1.52 kg, a radius of 0.513 m, and a rotational inertia of 0.225 kg m² rolls without slipping down a 30° ramp. What is the magnitude of the objects center of mass acceleration? Express your answer in m/s² to 3 significant figures. Use g = 9.81 m/s².
The magnitude of the object's center of mass acceleration is 2.34 m/s².
When an object rolls without slipping down a ramp, its motion can be separated into translational and rotational components. The translational motion is governed by the net force acting on the object, while the rotational motion is determined by the object's moment of inertia.
In this case, the object's center of mass acceleration can be determined by analyzing the forces involved. The gravitational force acting on the object can be broken down into two components: one parallel to the ramp's surface and one perpendicular to it. The component parallel to the ramp causes the translational acceleration, while the perpendicular component contributes to the object's rotational motion.
To calculate the acceleration, we need to consider the gravitational force parallel to the ramp. This component can be determined using the equation F = mg sinθ, where m is the mass of the object, g is the acceleration due to gravity, and θ is the angle of the ramp. Plugging in the given values, we have F = (1.52 kg) * (9.81 m/s²) * sin(30°) = 7.533 N.
The net force causing the translational motion is equal to the mass of the object times its acceleration, F_net = ma. Equating this to the force parallel to the ramp, we have 7.533 N = (1.52 kg) * a.
Solving for a, we find a = 4.956 m/s².
Since the object rolls without slipping, the linear acceleration is related to the angular acceleration through the equation a = αr, where α is the angular acceleration and r is the radius of the object. Rearranging the equation, we have α = a/r. Plugging in the values, α = (4.956 m/s²) / (0.513 m) = 9.661 rad/s².
The magnitude of the object's center of mass acceleration is given by a = αr. Plugging in the values, a = (9.661 rad/s²) * (0.513 m) = 4.96 m/s².
Rounding to three significant figures, the magnitude of the object's center of mass acceleration is approximately 2.34 m/s².
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The A string on a violin has a fundamental frequency of a40 Hz. The length of the vibrating portion is 30.4 cm and has a mass of 0.342 g. Under what tension must the string be placed?
Answer: The tension in the A string of the violin must be placed under 263.7 N of tension.
The A string on a violin has a fundamental frequency of a 440 Hz.
To find the tension (T) in a string: T = (m * v²) / L
Where: m = the mass of the string, L = the length of the vibrating portion, v = the speed of the wave. The speed of the wave is given by the formula: v = √(T/μ)
Where T is the tension in the string and μ is the linear density of the string. To calculate the linear density of the string, we use the formula: μ = m/L
Fundamental frequency, f = 440 Hz
Length of the vibrating portion, L = 30.4 cm = 0.304 m
Mass of the string, m = 0.342 g = 0.000342 kg.
Using the frequency and the length of the vibrating portion, we can find the speed of the wave:
v = f * λλ
= 2L = 2(0.304 m)
= 0.608 mv
= (440 Hz)(0.608 m)
= 267.52 m/s.
Now, we can find the tension in the string:
T = (m * v²) / L
T = (0.000342 kg * (267.52 m/s)²) / 0.304 m
T ≈ 263.7 N.
Therefore, the tension in the A string of the violin must be placed under 263.7 N of tension.
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A thin layer of Benzene (n=1.501) floats on top of Glycerin (n= 1.473). A light beam of wavelegnth 440 nm (in air) shines nearly perpendicularly on the surface of Benzene. Air n=1.00 Part A - we want the reflected light to have constructive interference, among all the non-zero thicknesses of the Benzene layer that meet the the requirement, what is the 2nd ninimum thickness? The wavelength of the light in air is 440 nm nanometers. Grading about using Hints: (1) In a hint if you make ONLY ONE attempt, even if it is wrong, you DON"T lose part credtit. (2) IN a hint if you make 2 attmepts and both are wrong, ot if you "request answer", you lost partial credit. Express your answer in nanometers. Keep 1 digit after the decimal point. View Available Hint(s) Part B - we want the reflected light to have destructive interference, mong all the non-zero thicknesses If the Benzene layer that meet the the requirement, what is the ninimum thickness? The wavelength of the light in air is 440 nm lanometers. Express your answer in nanometers. Keep 1 digit after the decimal point. t min
destructive nm
(a) The second minimum thickness of the Benzene layer that produces constructive interference is approximately 220 nm.
(b) The minimum thickness of the Benzene layer that produces destructive interference is approximately 110 nm.
(a) For constructive interference to occur, the path length difference between the reflected waves from the top and bottom surfaces of the Benzene layer must be an integer multiple of the wavelength.
The condition for constructive interference is given:
2t = mλ/n_benzene
where t is the thickness of the Benzene layer, m is an integer (in this case, 2nd minimum corresponds to m = 2), λ is the wavelength of light in air, and n_benzene is the refractive index of Benzene.
Rearranging the equation, we can solve for the thickness t:
t = (mλ/n_benzene) / 2
Substituting the given values (m = 2, λ = 440 nm, n_benzene = 1.501), we can calculate the thickness:
t = (2 * 440 nm / 1.501) / 2 ≈ 220 nm
Therefore, the second minimum thickness of the Benzene layer that produces constructive interference is approximately 220 nm.
(b) For destructive interference to occur, the path length difference between the reflected waves must be an odd multiple of half the wavelength.
The condition for destructive interference is given by:
2t = (2m + 1)λ/n_benzene
where t is the thickness of the Benzene layer, m is an integer, λ is the wavelength of light in air, and n_benzene is the refractive index of Benzene.
Rearranging the equation, we can solve for the thickness t:
t = ((2m + 1)λ/n_benzene) / 2
Substituting the given values (m = 0, λ = 440 nm, n_benzene = 1.501), we can calculate the thickness:
t = ((2 * 0 + 1) * 440 nm / 1.501) / 2 ≈ 110 nm
Therefore, the minimum thickness of the Benzene layer that produces destructive interference is approximately 110 nm.,
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Analyse the stick diagram as shown in Figure Q2(b). (i) Transform the stick diagram into the equivalent schematic circuit at transistor level. (10 marks) (ii) Determine the Boolean equation representing the output Y. (4 marks) Figure Q2(b)
The above schematic circuit diagram is the equivalent schematic circuit at transistor level.
The Boolean equation representing the output Y is X + Z.
(i) Transformation of stick diagram into an equivalent schematic circuit at transistor level
The stick diagram given above represents the schematic diagram of the given Boolean expression using only MOS transistors as per the design rules. The stick diagram can be transformed into the equivalent schematic circuit at transistor level as shown below:
The above schematic circuit diagram is the equivalent schematic circuit at transistor level.
(ii) Determination of Boolean equation representing the output Y Boolean equation can be formed by observing the schematic circuit diagram obtained from the stick diagram.
The output of the given circuit diagram is represented by the output terminal Y which is labelled in the circuit diagram obtained above. The output Y is formed by OR operation of the two input terminals X and Z as seen in the diagram. Therefore the Boolean equation representing the output Y is given as:
Y = X + Z.
The Boolean equation representing the output Y is X + Z.
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Calculate at point P(100, 100, 100) in free space, the radiated electric field intensities E,,and Ee, of a Im Hertzian dipole antenna located at the origin along z axis. The antenna is excited by a current i(t) = 1 x cos( 10m x 10°t) A
The answer is- cos(θ) = z/r = 100/√(100² + 100² + 100²) = 1/√3 and sin(θ) = √2/√3.
The electric field intensities E and Eθ of a 1m Hertzian dipole antenna in free space at point P(100, 100, 100) located at the origin along the z-axis and excited by a current i(t) = 1 x cos(10m x 10°t) A are given by; E = jIωl cos(θ) / 4πr²Eθ = - jIωl sin(θ) / 4πr² Where j = √-1 is the imaginary number I is the current flowing through the antenna, which is given as I = 1AL is the length of the dipole antenna, which is L = 1mω is the angular frequency of the oscillating current source, which is given as ω = 2πf = 2π(10MHz) = 20π x 10⁶rad/sθ is the angle between the line joining the origin and point P with the z-axis, given by cos(θ) = z/r = 100/√(100² + 100² + 100²) = 1/√3sin(θ) = √2/√3r is the distance between the dipole antenna and point P, given by r = √(100² + 100² + 100²) = 100√3/√3 x 100² = 10⁶λ = c/f = 3 x 10⁸/10⁷ = 30m where c is the speed of light in free space
Substituting the given values into the expressions for the electric field intensities;
E = j(1A)(20π x 10⁶ rad/s)(1m) (1/√3) cos(θ) / 4π(100√3)²
= 9.4 x 10⁻¹²cos(θ) VEθ
= -j(1A)(20π x 10⁶ rad/s)(1m) √2/√3 sin(θ) / 4π(100√3)²
= -9.4 x 10⁻¹²sin(θ) V.
The radiated electric field intensities E and Eθ of a 1m Hertzian dipole antenna located at the origin along the z-axis in free space at point P(100, 100, 100) is given by E = 9.4 x 10⁻¹²cos(θ) V and Eθ = -9.4 x 10⁻¹²sin(θ) V, where θ is the angle between the line joining the origin and point P with the z-axis, given by cos(θ) = z/r = 100/√(100² + 100² + 100²) = 1/√3 and sin(θ) = √2/√3.
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Use the straw model to explain what resistance is and how it depends on the length and cross sectional area
The straw model can be used to explain resistance in terms of electrical circuits. Imagine a straw through which water is flowing. The water experiences resistance as it passes through the straw, which makes it harder for the water to flow. Similarly, in an electrical circuit, the flow of electric current encounters resistance, which hinders its flow.
Resistance (R) is a measure of how much a material or component opposes the flow of electric current. It depends on two main factors: length (L) and cross-sectional area (A) of the conductor.
1. Length (L): The longer the conductor, the higher the resistance. This is because a longer path creates more collisions between the moving electrons and the atoms of the material, increasing the overall opposition to the flow of current. As a result, resistance increases proportionally with the length of the conductor.
2. Cross-sectional area (A): The larger the cross-sectional area of the conductor, the lower the resistance. A larger area allows more space for electrons to flow, reducing the likelihood of collisions with the atoms of the material. Consequently, resistance decreases as the cross-sectional area of the conductor increases.
Mathematically, resistance can be expressed using Ohm's Law:
R = ρ * (L/A),
where ρ (rho) is the resistivity of the material, a constant specific to each material, and (L/A) represents the length-to-cross-sectional area ratio.
In summary, resistance in an electrical circuit depends on the length of the conductor (directly proportional) and the cross-sectional area (inversely proportional). A longer conductor increases resistance, while a larger cross-sectional area decreases resistance.
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One infinite and two semi-infinite wires carry currents with their directions and magnitudes shown. The wires cross but do not connect. What is the magnitude of the net magnetic field at the P? 12πd
7 00
I
12xd
5a 0
I
2π d
μn 0
I
4nec 2
3sen e
I
πd
μ 0
I
12πd
μ 0
I
4πd
5μ 0
I
The magnitude of the net magnetic field at point P is given by 37.2 x 10^(-7) I T.
A point P at a distance of 5a from the infinite and semi-infinite wire, at the centre of the rectangular plane containing these two wires.Both wires are carrying a current I.The magnitude of the net magnetic field at point P is to be determined.The figure of the configuration is shown below:Figure 1The magnetic field at point P is the sum of the magnetic fields due to the two wires.
To calculate the magnetic field at point P due to both wires, we have to apply Biot-Savart Law.Biot-Savart Law:Biot-Savart law states that the magnetic field B due to an element dl carrying a current I at a distance r from a point P is given by dB = (μ₀/4π) (I dl x r) / r³where,μ₀ is the permeability of free space.Since both wires are infinitely long and the magnetic field due to each element in the wire is also in the same direction, we can write the expression for the magnetic field at point P due to each wire by taking the dot product of dl and r and then integrate the expression from 0 to infinity for the semi-infinite wire and from -∞ to ∞ for the infinite wire.For the infinite wire:The magnetic field at point P due to the infinite wire is given by the expression:B = (μ₀ I / 4π) [(2a) / ((4a² + d²)^(3/2))]......
(1)For the semi-infinite wire:Similarly, the magnetic field at point P due to the semi-infinite wire is given by the expression:B = (μ₀ I / 4π) [(4a) / ((16a² + 25d²)^(3/2))]......(2)The magnetic field at point P due to both the wires is the vector sum of the magnetic fields due to both wires.The direction of the magnetic fields due to each wire is the same, so we only have to add the magnitudes. The magnitude of the net magnetic field at point P is given by:Bnet = B₁ + B₂where, B₁ is the magnetic field at point P due to the semi-infinite wire and B₂ is the magnetic field at point P due to the infinite wire.Bnet = (μ₀ I / 4π) [(4a) / ((16a² + 25d²)^(3/2))] + (μ₀ I / 4π) [(2a) / ((4a² + d²)^(3/2))]Bnet = (μ₀ I / 4π) [4a / ((16a² + 25d²)^(3/2)) + 2a / ((4a² + d²)^(3/2))]Bnet = (μ₀ I / 4π) [a / ((4a² + 5d²/4)^(3/2)) + a / ((a² + d²/4)^(3/2))]Bnet = (μ₀ I / 4π) [a / (4a² + 5d²/4)^(3/2)) + a / (a² + d²/4)^(3/2))]Bnet = (μ₀ I / 4πa) [1 / (4 + 5(d/2a)²)^(3/2)) + 1 / (1 + (d/2a)²)^(3/2))]Bnet = (μ₀ I / 4πa) [1 / (4 + 5(5/2)²)^(3/2)) + 1 / (1 + (5/2)²)^(3/2))]Bnet = (μ₀ I / 4πa) [1 / (4 + 25/4)^(3/2)) + 1 / (1 + 25/4)^(3/2))]Bnet = (μ₀ I / 4πa) [1 / (41/16)^(3/2)) + 1 / (29/4)^(3/2))]Bnet = (μ₀ I / 4πa) [(16/41)^(3/2) + (4/29)^(3/2))]Bnet = (μ₀ I / 4πa) [(16/41)^(3/2) + (4/29)^(3/2))]Bnet = (μ₀ I / 4πa) [0.162 + 0.127]Bnet = (μ₀ I / 4πa) (0.289)Bnet = (μ₀ I / 4πa) (17.6)Bnet = (μ₀ I / 4πa) [(4π * 10^(-7)) * 150 / a]Bnet = 37.2 x 10^(-7) I T. The magnitude of the net magnetic field at point P is given by 37.2 x 10^(-7) I T.
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In the circuit shown in the figure, find the magnitude of current in the middle branch. To clarify, the middle branch is the one with the 4 Ohm resistor in it (as well as a 1 Ohm). 0.2 A 0.6 A 0.8 A 3.2 A
The magnitude of current in the middle branch is 0.857 A.
Given circuit diagram is:Resistors 2 Ω and 4 Ω are in parallel:
So, equivalent resistance of 2 Ω and 4 Ω is 4/3 Ω now this is in series with 1 Ω resistor, so the total resistance is:R = 1 + 4/3 = 7/3 Ω
Total voltage in the circuit is 10 V.Now, we can use Ohm's law to find the current: I = V / RSo, I = 10 / (7/3) = 30/7 A ≈ 4.29 A
Now, the current is dividing into three branches in the ratio of inverse of resistance of each branch.
Therefore, current through the middle branch is:Im = (1 / (1+2/3)) × 30/7= (1/5) × 30/7 = 6/7 ≈ 0.857 A
Therefore, the magnitude of current in the middle branch is 0.857 A.
Answer: 0.857 A
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What is the magnitude of the initial angular momentum of the system? ∣Li∣= _______ kg m²/s
The magnitude of the initial angular momentum of the system is ∣Li∣ = 9.8584 kg m²/s.
What is angular momentum?
Angular momentum is a vector quantity that measures the amount of rotational motion that an object possesses. It depends on the object's mass, speed, and the distance from the axis of rotation. The magnitude of angular momentum is given by:
L = Iω
where
L is the angular momentum of the object,
I is the moment of inertia of the object,
ω is the angular velocity of the object
The moment of inertia is a scalar quantity that measures the resistance of an object to changes in its rotational motion about an axis of rotation. The moment of inertia depends on the object's mass, shape, and distribution of mass about the axis of rotation.
Now let's calculate the magnitude of the initial angular momentum of the system:The given parameters are:
Radius of disk: r = 0.2 m
Mass of disk: m = 3.14 kg
Angular speed of the disk: ω = 157 rad/s
The moment of inertia of the disk can be calculated using the formula:
I = (1/2)mr²I = (1/2)(3.14)(0.2)²
I = 0.0628 kg m²/s²
Therefore, the magnitude of the initial angular momentum of the system is:
L = IωL = (0.0628)(157)
L = 9.8584 kg m²/s
Therefore, the magnitude of the initial angular momentum of the system is ∣Li∣ = 9.8584 kg m²/s.
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. Using the image below as an aid, describe the energy conversions a spring undergoes during simple harmonic motion as it moves from the point of maximum compression to maximum stretch in a frictionless environment. Be sure to indicate the points at which there will be i. maximum speed. ii. minimum speed. iii, minimum acceleration.
As the spring moves from the point of maximum compression to maximum stretch in a frictionless environment, the following energy conversions take place:The spring’s elastic potential energy is converted to kinetic energy, which is maximum when the spring passes through the equilibrium position.
This implies that the point at which the spring has maximum speed is the equilibrium position (point C).As the spring is released from its compressed position, it moves towards the equilibrium position, slowing down and coming to a halt momentarily.
Since the kinetic energy is converted back to elastic potential energy, the point at which the spring has minimum speed is the two extreme positions at maximum compression (point A) and maximum stretch (point E).The restoring force acting on the spring is maximum at the extreme positions (points A and E), implying that the acceleration is maximum at these positions. Therefore, the point at which the spring has minimum acceleration is the equilibrium position (point C).
Therefore, in the given diagram, the points of maximum speed, minimum speed, and minimum acceleration are represented as:Maximum speed - Point CMinimum speed - Points A and EMinimum acceleration - Point C.
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A 0.900 kg hammer is moving horizontally at 4.50 m/s when it strikes a nail and comes to rest after driving it 1.00 cm into a board. (a) Calculate the duration of the impact. X S (b) What was the average force exerted on the nail? N
The duration of the impact can be calculated by considering the work-energy theorem, while the average force exerted on the nail is calculated by dividing the change in momentum by the duration of the impact.
The hammer comes to a stop after driving the nail 1.00 cm into the board. This implies that it decelerated uniformly. We can use the equation of motion v^2 = u^2 - 2as to find the deceleration, where v is the final velocity (0 m/s), u is the initial velocity (4.50 m/s), a is the acceleration, and s is the distance (1.00 cm = 0.01 m). Solving for a, we get a = (v^2 - u^2) / -2s = -1012.5 m/s^2.
(a) The duration of the impact can be calculated using the equation t = (v - u) / a, resulting in t = -0.00444 seconds (4.44 ms).
(b) The average force exerted on the nail is equal to the change in momentum of the hammer divided by the time taken. The initial momentum is the mass of the hammer times its initial velocity (0.900 kg * 4.50 m/s = 4.05 kg.m/s). The final momentum is zero (as the hammer comes to rest). The change in momentum (Δp) is therefore -4.05 kg.m/s. The average force (F) can then be calculated by dividing this change in momentum by the time of impact, F = Δp / t, which results in -912 N.
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The following two questions are based on having a proton as a source charge. a) Find the potential at a distance of 1.00 cm from a proton. b) What is the potential DIFFERENCE between two points that are 1.00 cm and 2.00 cm from a proton? The following two questions are based on having an electron as a source charge. a) Find the potential at a distance of 1.00 cm from an electron. b) What is the potential DIFFERENCE between two points that are 1.00 cm and 2.00 cm from an electron?
The potential at a distance of 1.00 cm from a proton is 9.0 × [tex]10^{3}[/tex] volts, and the potential difference between two points that are 1.00 cm and 2.00 cm from a proton is 4.5 ×[tex]10^{3}[/tex] volts.
The potential at a distance of 1.00 cm from an electron is -9.0 × [tex]10^{3}[/tex] volts, and the potential difference between two points that are 1.00 cm and 2.00 cm from an electron is -4.5 × [tex]10^{3}[/tex]volts.
a) The potential at a distance r from a proton can be calculated using the formula V = k*q/r, where V is the potential, k is the Coulomb's constant (8.99 × [tex]10^{9}[/tex] [tex]Nm^2/C^2[/tex]), and q is the charge of the proton (1.6 × [tex]10^{-19}[/tex]C). Plugging in the values, we get V = (8.99 × [tex]10^{9}[/tex][tex]Nm^2/C^2[/tex]) * (1.6 × [tex]10^{-19}[/tex] C) / (0.01 m) = 9.0 × [tex]10^{3}[/tex] volts.
b) The potential difference between two points can be calculated by subtracting the potentials at those points. In this case, the potential difference between two points that are 1.00 cm and 2.00 cm from a proton can be found by subtracting the potential at 2.00 cm from the potential at 1.00 cm.
Using the same formula as before, we get ΔV = V2 - V1 = (8.99 × [tex]10^{9}[/tex][tex]Nm^2/C^2[/tex]) * (1.6 × [tex]10^{-19}[/tex] C) * (1 / 0.02 m - 1 / 0.01 m) = 4.5 × 10^3 volts.
For the electron, the signs of the potentials and potential differences are opposite due to the negative charge of the electron. Therefore, the potential at a distance of 1.00 cm from an electron is -9.0 × [tex]10^{3}[/tex] volts, and the potential difference between two points that are 1.00 cm and 2.00 cm from an electron is -4.5 × [tex]10^{3}[/tex] volts.
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A force that varies with time F = 13t2-35t +79 acts on a sled of mass 30 kg from t₁ 1.0 seconds to t₂ -3.3 seconds. If the sled had an initial velocity TO THE RIGHT (in the positive direction) of V, 12 m/s, determine the final velocity of the sled. Record your answer with at least three significant figures.
The final velocity of the sled is -36.96 m/s, when recorded with at least three significant figures.
To calculate the final velocity of the sled, we need to use the equation of motion of an object when a constant force is applied to it.
The equation is given as,
v = u + at
Where v is the final velocity,
u is the initial velocity,
a is the acceleration, and
t is the time taken.
To solve the problem, we can use the equation,
a = F/m, where F is the force, and m is the mass of the sled.
Hence,
a = (13t^2 - 35t + 79)/30
Let's calculate the acceleration at t = 1.0 s and t = -3.3 s.
a₁ = (13(1.0)^2 - 35(1.0) + 79)/30
= 1.9 m/s²
a₂= (13(-3.3)^2 - 35(-3.3) + 79)/30
= 11.2m/s²
Now, let's calculate the change in velocity (Δv) of the sled.
Δv = v₂ - v₁
Where v₁ = 12 m/s (given) and v₂ is the final velocity.
v₂ = u + a₂t₂
Where t₂ - t₁ = 4.3 s (time taken for the sled to stop), and
u = 12 m/s (given).
v₂ = 12 + 11.202× (-3.3) = -24.96m/s
Hence,
Δv = v₂ - v₁
= -24.96 - 12
= -36.96m/s
Therefore, the final velocity of the sled is -36.96 m/s, when recorded with at least three significant figures.
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