Question 1

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 answer of An air-track cart is attached to a...

Question 2

The position of an air-track cart that is oscillating on a spring is given by the equation x = (12.4 cm)cos[(6.35 s^-1)t].At what value of t after t = 0.00 s is the cart first located at x = 8.47 cm?

Accepted Answer

We need to solve for t when x = 8.47 cm:

8.47 = 12.4cos[(6.35 s^-1)t]

cos[(6.35 s^-1)t] = 8.47/12.4 = 0.683

Using the inverse cosine function on both sides:

(6.35 s^-1)t = cos^-1(0.683)

t = cos^-1(0.683)/6.35 s^-1

t ≈ 0.129 s

Therefore, the correct choice is (C).

Question 3

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?

Accepted Answer

The answer of A point on the string of a...

Question 4

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?

Accepted Answer

The answer of A sewing machine needle moves in simple...

Question 5

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

Accepted Answer

The period of oscillation (T) is the reciprocal of the frequency (f), so T = 1/f. Given the frequency is 32.8 kHz, or 32,800 Hz, the period T = 1/32,800 = 30.4878 microseconds, which rounds to 30.5 µs.

Question 6

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now accelerates downward uniformly at 2 m/s²,what is true about the periods of these two systems?

Accepted Answer

The period of a simple pendulum depends on the acceleration due to gravity (g). If the elevator accelerates downward, the effective g decreases, leading to an increase in the pendulum's period. The period of an ideal spring-mass system, however, depends on the mass and the spring constant, not on g, so it remains unchanged.

Question 7

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?

Accepted Answer

The answer of A sewing machine needle moves up and...

Question 8

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

Accepted Answer

The answer of If a pendulum makes 12 complete swings...

Question 9

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How would the period of the pendulum change if the supporting chain were to break,putting the elevator into freefall?

Accepted Answer

When the elevator is in freefall, both the pendulum and the elevator are in a state of freefall, experiencing the same acceleration due to gravity. This means there is no force acting on the pendulum to cause it to swing, as it is effectively in a zero-gravity environment relative to the elevator. Therefore, the pendulum would not swing at all, making its period infinite in this context.

Question 10

If your heart is beating at 76.0 beats per minute,what is the frequency of your heart's oscillations in hertz?

Accepted Answer

The frequency in hertz (Hz) is the number of cycles (in this case, heartbeats) per second. To convert beats per minute to hertz, divide the number of beats by 60 (since there are 60 seconds in a minute). Therefore, 76.0 beats per minute is 76.0/60 = 1.27 Hz.

Question 11

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now moves downward at a uniform 2 m/s,what is true about the periods of these two systems?

Accepted Answer

The period of a simple pendulum depends on the length of the pendulum and the acceleration due to gravity, neither of which changes in an elevator moving downward at a constant velocity. The period of an object on an ideal spring depends on the mass of the object and the spring constant, neither of which is affected by the elevator's constant downward velocity. Therefore, both periods remain the same.

Question 12

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves downward with constant acceleration?

Accepted Answer

The answer of A pendulum of length L is suspended...

Question 13

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves upward with constant velocity?

Accepted Answer

The period of a pendulum depends only on its length and the acceleration due to gravity (g), not on the velocity of the elevator. Since the elevator is moving with constant velocity, there is no acceleration acting on the pendulum, so its period remains constant.

Question 14

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 oscillation?

Accepted Answer

The answer of If a floating log is seen to...

Question 15

A guitar string is set into vibration with a frequency of 512 Hz.How many oscillations does it undergo each minute?

Accepted Answer

To find the number of oscillations per minute, we have to first convert the frequency from Hz to oscillations per second, and then multiply by 60 to get the oscillations per minute.

512 Hz = 512 oscillations per second

512 x 60 = 30,720 oscillations per minute

Therefore, the closest option to this is A) 30,700 oscillations per minute.

Question 16

A leaky faucet drips 40 times in $30.0 \mathrm {~s} .$ What is the frequency of the dripping?

Accepted Answer

The frequency of the dripping is calculated by dividing the number of drips by the time in seconds. So, $ \frac{40 \, \text{drips}}{30.0 \, \text{s}} = 1.33 \, \text{Hz} $, which rounds to 1.3 Hz.

Question 17

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?

Accepted Answer

The answer of The position of a cart that is...

Question 18

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

Accepted Answer

The maximum acceleration in simple harmonic motion is given by $a_{max} = \omega^2 x_{max}$, where $\omega$ is the angular frequency and $x_{max}$ is the maximum displacement. Since $\omega = 2\pi f$, doubling the frequency $f$ quadruples the square of the angular frequency $\omega^2$, thus quadrupling the maximum acceleration.

Question 19

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?

Accepted Answer

The answer of The position of an object that is...

Question 20

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves upward with constant acceleration?

Accepted Answer

When the elevator moves upward, the tension in the pendulum thread is greater than its weight, resulting in a shorter effective length of the pendulum. Thus, the period of the pendulum will decrease. This can be demonstrated mathematically using the formula for the period of a simple pendulum: 
T = 2π√(L/g) 
Where L is the length of the pendulum and g is the acceleration due to gravity. 
As the elevator moves upward with constant acceleration a, the effective value of g in the elevator frame of reference is:
g' = g + a 
Therefore, the period of the pendulum in the elevator frame of reference is:
T' = 2π√(L/(g+a))
Expanding this and simplifying yields:
T' = T/(1 + (a/g))
Since a is positive, this expression is less than T, implying that the period of the pendulum decreases when the elevator is moving upward.

Quiz 20: Vibrational Motion

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?

The position of an air-track cart that is oscillating on a spring is given by the equation x = (12.4 cm) cos[(6.35 s^-1) t].At what value of t after t = 0.00 s is the cart first located at x = 8.47 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?

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

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now accelerates downward uniformly at 2 m/s²,what is true about the periods of these two systems?

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?

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

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How would the period of the pendulum change if the supporting chain were to break,putting the elevator into freefall?

If your heart is beating at 76.0 beats per minute,what is the frequency of your heart's oscillations in hertz?

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now moves downward at a uniform 2 m/s,what is true about the periods of these two systems?

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves downward with constant acceleration?

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves upward with constant velocity?

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 oscillation?

A guitar string is set into vibration with a frequency of 512 Hz.How many oscillations does it undergo each minute?

A leaky faucet drips 40 times in $30.0 \mathrm {~s} .$ What is the frequency of the dripping?

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?

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 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 pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves upward with constant acceleration?

Quiz 20: Vibrational Motion

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?

The position of an air-track cart that is oscillating on a spring is given by the equation x = (12.4 cm) cos[(6.35 s-1) t].At what value of t after t = 0.00 s is the cart first located at x = 8.47 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?

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

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now accelerates downward uniformly at 2 m/s2,what is true about the periods of these two systems?

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?

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

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How would the period of the pendulum change if the supporting chain were to break,putting the elevator into freefall?

If your heart is beating at 76.0 beats per minute,what is the frequency of your heart's oscillations in hertz?

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now moves downward at a uniform 2 m/s,what is true about the periods of these two systems?

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves downward with constant acceleration?

A pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves upward with constant velocity?

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 oscillation?

A guitar string is set into vibration with a frequency of 512 Hz.How many oscillations does it undergo each minute?

A leaky faucet drips 40 times in 30.0 s.30.0 \mathrm {~s} .30.0 s. What is the frequency of the dripping?

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?

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 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 pendulum of length L is suspended from the ceiling of an elevator.When the elevator is at rest the period of the pendulum is T.How does the period of the pendulum change when the elevator moves upward with constant acceleration?

The position of an air-track cart that is oscillating on a spring is given by the equation x = (12.4 cm) cos[(6.35 s^-1) t].At what value of t after t = 0.00 s is the cart first located at x = 8.47 cm?

A simple pendulum and an object oscillating on an ideal spring both have period T in an elevator at rest.If the elevator now accelerates downward uniformly at 2 m/s²,what is true about the periods of these two systems?

A leaky faucet drips 40 times in $30.0 \mathrm {~s} .$ What is the frequency of the dripping?

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?

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?