Question 1

The position of an air-track cart that is oscillating on a spring is given by the equation  x=(12.4 cm)  
At what value of  t  after  t=0.00 s is the cart first located at  x=8.47 cm?

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Question 2

An object is oscillating on a spring with a period of  4.60 s. At time  t=0.00 s the object has zero speed and is at  x=8.30 cm. What is the acceleration of the object at  t=2.50 s?

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Question 3

If the amplitude of the motion of a simple harmonic oscillator is doubled, by what factor does the maximum speed of the oscillator change?

Accepted Answer

The maximum speed of a simple harmonic oscillator is directly proportional to the amplitude. Therefore, if the amplitude is doubled, the maximum speed also doubles.

Question 4

The quartz crystal in a digital watch has a frequency of 32.8 kHz. What is its period of oscillation?

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Question 5

If the frequency of a system undergoing simple harmonic motion doubles, by what factor does the maximum value of acceleration change?

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Question 6

The position of an object that is oscillating on an ideal spring is given by  x=(17.4 cm)  
Write an expression for the velocity of the particle as a function of time using the sine function.

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Question 7

An object undergoing simple harmonic motion has a maximum displacement of  6.2 m at  t=0.00 s. If the angular frequency of oscillation is  1.6 rad/s, what is the object's displacement when  t=3.5 s?

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Question 8

If a floating log is seen to bob up and down 15 times in 1.0 min as waves pass by you,
what are the frequency and period of the wave?

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Question 9

A ball is oscillating on an ideal spring with an amplitude of  8.3 cm and a period of  4.6 s. Write an expression for its position,  x , as a function of time  t , if  x  is equal to  8.3 cm at  t=   0.0 s. Use the cosine function.

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Question 10

If the frequency of the motion of a simple harmonic oscillator is doubled, by what factor does the maximum speed of the oscillator change?

Accepted Answer

The maximum speed of a simple harmonic oscillator is directly proportional to the frequency. Therefore, if the frequency is doubled, the maximum speed also doubles.

Question 11

A package is oscillating on a spring scale with a period of  4.60 s. At time  t=0.00 s the package has zero speed and is at  x=8.30 cm. At what time after  t=0.00 s will the package first be at  x=4.15 cm?

Accepted Answer

The package is undergoing simple harmonic motion. The time to reach half the amplitude (from 8.30 cm to 4.15 cm) is a quarter of the period, which is 4.60 s / 4 = 1.15 s.

Question 12

A point on the string of a violin moves up and down in simple harmonic motion with an
amplitude of 1.24 mm and a frequency of 875 Hz.
(a) What is the maximum speed of that point in SI units?
(b) What is the maximum acceleration of the point in SI units?

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Question 13

An air-track cart is attached to a spring and completes one oscillation every  5.67 s in simple harmonic motion. At time  t=0.00 s the cart is released at the position  x=+0.250 m. What is the position of the cart when  t=29.6 s ?

Accepted Answer

The position of the cart in simple harmonic motion can be determined using the formula $x = A \cos(2\pi \frac{t}{T} + \phi)$, where $A$ is the amplitude, $T$ is the period, $t$ is the time, and $\phi$ is the phase constant. Given that the cart completes one oscillation every 5.67 s, the period $T = 5.67$ s. The cart is released at $x = +0.250$ m at $t = 0.00$ s, which means $A = 0.250$ m and $\phi = 0$ (since it starts at the maximum displacement). To find the position at $t = 29.6$ s, we substitute these values into the formula:$$x = 0.250 \cos\left(2\pi \frac{29.6}{5.67}\right)$$The fraction $\frac{29.6}{5.67}$ gives the number of cycles completed, and since we're interested in the position at a specific time, we need to calculate the cosine of this fraction times $2\pi$. The exact value will depend on the calculator, but the key is recognizing that the position repeats every period, and 29.6 s is equivalent to some whole number of periods plus a fraction of a period. Calculating the exact position requires a calculator for the cosine value, but the principle is that the position is a function of the cosine of the time divided by the period, multiplied by $2\pi$, which gives us the phase of the motion at that time. Since the question is about the qualitative understanding and calculation of the position, the exact numerical calculation would show that the position is closest to the value given in option B, $x = 0.0461$ m, based on the periodic nature of the cosine function and the given parameters.

Question 14

If a pendulum makes 12 complete swings in 8.0 s, what are its (a) frequency and (b)
period?

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Question 15

The position of an object that is oscillating on a spring is given by the equation x = (18.3
cm) cos[(2.35 s-1)t]. What are the (a) frequency, (b) amplitude, and (c) period of this
motion?

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Question 16

A sewing machine needle moves up and down in simple harmonic motion with an
amplitude of 1.27 cm and a frequency of 2.55 Hz. What are the (a) maximum speed and
(b) maximum acceleration of the tip of the needle?

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Question 17

The position of an object that is oscillating on an ideal spring is given by  x=(17.4 cm)  
Write an expression for the acceleration of the particle as a function of time using the cosine function.

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Question 18

An object oscillates such that its position  x  as a function of time  t  obeys the equation   where  t  is in seconds.
(a) In one period, what total distance does the object move?
(b) What is the frequency of the motion?
(c) What is the position of the object when  t=1.00 s?

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Question 19

A sewing machine needle moves in simple harmonic motion with a frequency of 2.5 Hz
and an amplitude of 1.27 cm.
(a) How long does it take the tip of the needle to move from the highest point to the lowest
point in its travel?
(b) How long does it take the needle tip to travel a total distance of 11.43 cm?

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Question 20

The position of a cart that is oscillating on a spring is given by the equation x = (12.3 cm)
cos[(1.26 s-1)t]. When t = 0.805 s, what are the (a) velocity and (b) acceleration of the cart?

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Quiz 11: Oscillations and Waves

The position of an air-track cart that is oscillating on a spring is given by the equation x=(12.4 cm) At what value of t after t=0.00 s is the cart first located at x=8.47 cm?

An object is oscillating on a spring with a period of 4.60 s. At time t=0.00 s the object has zero speed and is at x=8.30 cm. What is the acceleration of the object at t=2.50 s?

If the amplitude of the motion of a simple harmonic oscillator is doubled, by what factor does the maximum speed of the oscillator change?

The quartz crystal in a digital watch has a frequency of 32.8 kHz. What is its period of oscillation?

If the frequency of a system undergoing simple harmonic motion doubles, by what factor does the maximum value of acceleration change?

The position of an object that is oscillating on an ideal spring is given by x=(17.4 cm) Write an expression for the velocity of the particle as a function of time using the sine function.

An object undergoing simple harmonic motion has a maximum displacement of 6.2 m at t=0.00 s. If the angular frequency of oscillation is 1.6 rad/s, what is the object's displacement when t=3.5 s?

If a floating log is seen to bob up and down 15 times in 1.0 min as waves pass by you, what are the frequency and period of the wave?

A ball is oscillating on an ideal spring with an amplitude of 8.3 cm and a period of 4.6 s. Write an expression for its position, x , as a function of time t , if x is equal to 8.3 cm at t= 0.0 s. Use the cosine function.

If the frequency of the motion of a simple harmonic oscillator is doubled, by what factor does the maximum speed of the oscillator change?

A package is oscillating on a spring scale with a period of 4.60 s. At time t=0.00 s the package has zero speed and is at x=8.30 cm. At what time after t=0.00 s will the package first be at x=4.15 cm?

A point on the string of a violin moves up and down in simple harmonic motion with an amplitude of 1.24 mm and a frequency of 875 Hz. (a) What is the maximum speed of that point in SI units? (b) What is the maximum acceleration of the point in SI units?

An air-track cart is attached to a spring and completes one oscillation every 5.67 s in simple harmonic motion. At time t=0.00 s the cart is released at the position x=+0.250 m. What is the position of the cart when t=29.6 s ?

If a pendulum makes 12 complete swings in 8.0 s, what are its (a) frequency and (b) period?

The position of an object that is oscillating on a spring is given by the equation x = (18.3 cm) cos[(2.35 s-1)t]. What are the (a) frequency, (b) amplitude, and (c) period of this motion?

A sewing machine needle moves up and down in simple harmonic motion with an amplitude of 1.27 cm and a frequency of 2.55 Hz. What are the (a) maximum speed and (b) maximum acceleration of the tip of the needle?

The position of an object that is oscillating on an ideal spring is given by x=(17.4 cm) Write an expression for the acceleration of the particle as a function of time using the cosine function.

An object oscillates such that its position x as a function of time t obeys the equation where t is in seconds. (a) In one period, what total distance does the object move? (b) What is the frequency of the motion? (c) What is the position of the object when t=1.00 s?

The position of a cart that is oscillating on a spring is given by the equation x = (12.3 cm) cos[(1.26 s-1)t]. When t = 0.805 s, what are the (a) velocity and (b) acceleration of the cart?