Question 1

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46 s^-1)t]. What is the angular frequency for this motion?

Accepted Answer

The equation given for the position of the mass is x = (17.4 cm) cos[(5.46 rad/s)t], where the angular frequency ω = 5.46 rad/s. This can be seen by comparing the given equation to the standard form for simple harmonic motion, x = A cos(ωt), where A is the amplitude and ω is the angular frequency. Therefore, the answer is choice B, 5.46 rad/s.

Question 2

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. In any 4.60 s time interval, how much time is the mass more than 4.15 cm away from x = 0 cm?

Accepted Answer

The mass is in simple harmonic motion with an amplitude of 8.30 cm. It is more than 4.15 cm away from x = 0 cm for half of the period, which is 2.30 s. However, since it is symmetric, it is more than 4.15 cm away for 3.07 s in total during one full period.

Question 3

When a mass of 0.350 kg is attached to a vertical spring and lowered slowly, the spring stretches 12.0 cm. The mass is now displaced from its equilibrium position and undergoes simple harmonic oscillations. What is the period of the oscillations?

Accepted Answer

The spring constant k can be found using Hooke's Law: F = kx, where F = mg. Then, the period T of oscillation is given by T = 2π√(m/k).

Question 4

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 is the maximum acceleration of the needle?

Accepted Answer

The maximum acceleration of an object in simple harmonic motion can be found using the formula a_max = ω^2*A, where ω is the angular frequency (2πf) and A is the amplitude. 
In this case, ω = 2π(2.55 Hz) = 16.06 rad/s and A = 1.27 cm = 0.0127 m. 
Thus, a_max = (16.06 rad/s)^2 * 0.0127 m ≈ 3.26 m/s^2 = 326 cm/s^2. 
Therefore, the correct answer is D: 326 cm/s^2.

Question 5

A 92-kg man climbs onto a car with worn out shock absorbers and this causes the car to drop down 4.5 cm. As he drives along, he hits a bump, and this starts the car oscillating at an angular frequency of 4.52 rad/s. What is the mass of the car?

Accepted Answer

The correct answer is A because the mass of the car can be calculated using the formula: mass = (force of gravity * distance traveled) / (angular frequency^2 * amplitude). Plugging in the given values, we get: mass = (92 kg * 9.8 m/s^2 * 0.045 m) / (4.52 rad/s)^2 * 0.045 m) = 890 kg.

Question 6

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. How long does it take it to travel 11.43 cm?

Accepted Answer

The time for one complete cycle is the period, T = 1/frequency = 1/2.55 Hz = 0.392 s. The needle travels 4 times the amplitude in one cycle (up and down twice), so it travels 4 * 1.27 cm = 5.08 cm per cycle. To travel 11.43 cm, it takes 11.43 cm / 5.08 cm per cycle = 2.25 cycles. Therefore, the time taken is 2.25 cycles * 0.392 s/cycle = 0.882 s.

Question 7

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is its speed at t = 2.50 s?

Accepted Answer

The answer of A mass is oscillating on a spring...

Question 8

A mass of 0.150 kg is attached to a spring with a force constant of 3.58 N/m and undergoes simple harmonic oscillations. What is the period of the oscillations?

Accepted Answer

The answer of A mass of 0.150 kg is attached...

Question 9

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s²)t]. What is the acceleration of the mass when t = 0.825 s?

Accepted Answer

The acceleration of the mass is given by the second derivative of the position function with respect to time. The second derivative of cos(t) is -cos(t), so the acceleration of the mass is given by:a = -Aω^2cos(ωt)where A is the amplitude of the oscillation, ω is the angular frequency, and t is the time.At t = 0.825 s, the acceleration of the mass is:a = -(12.3 cm)(1.26 s^-2)^2cos[(1.26 s^-2)(0.825 s)] = -9.89 cm/s^2Therefore, the correct answer is B.

Question 10

In a supermarket, you place a 5 lb (22.3 N) bag of oranges on a scale. The scale starts to oscillate at 2.70 Hz. What is the force constant of the scale?

Accepted Answer

The force constant (k) can be calculated using the formula for the frequency of oscillation: f = (1/2π) * √(k/m). Rearranging gives k = (2πf)² * m. Using m = 22.3 N / 9.8 m/s² and f = 2.70 Hz, we find k ≈ 654 N/m.

Question 11

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is the value of t the first time after t = 0 s that the mass is at x = 4.15 cm?

Accepted Answer

The answer of A mass is oscillating on a spring...

Question 12

An air-track cart is attached to a spring and completes one oscillation every 5.67 s. At t = 0 s the cart is released at a distance of 0.250 m from its equilibrium position. 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 13

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is its acceleration at t = 2.50 s?

Accepted Answer

We can use the equation for the position of a mass on a spring:
x = Acos(ωt + φ)
where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle.

We are told that the period is 4.60 s, which means the angular frequency is:
ω = 2π/T = 2π/4.60 s = 1.36 rad/s

We are also told that the mass is at x = 8.30 cm and has zero speed at t = 0 s, which means it starts at the equilibrium position and has no phase angle:
x = Acos(ωt)
8.30 cm = A*cos(0)
A = 8.30 cm

Now we can find the acceleration at t = 2.50 s using the equation for acceleration:
a = -ω^2Acos(ωt)
a = -1.36^2*8.30*cos(1.36*2.50)
a = -14.9 cm/s^2

Therefore, the answer is D: 14.9 cm/s^2.

Question 14

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s^-1)t]. What is the speed of the mass when t = 0.805 s?

Accepted Answer

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

Question 15

A mass of 0.250 kg is attached to a spring and undergoes simple harmonic oscillations with a period of 0.640 s. What is the force constant of the spring?

Accepted Answer

The period of simple harmonic motion is given by the equation:

T = 2π√(m/k)

where T is the period, m is the mass, and k is the force constant of the spring.

Rearranging the equation, we get:

k = (4π^2m) / T^2

Plugging in the given values, we get:

k = (4π^2 * 0.250 kg) / (0.640 s)^2

k = 24.1 N/m

Therefore, the force constant of the spring is 24.1 N/m, which corresponds to choice C.

Question 16

When a mass of 0.350 kg is attached to a vertical spring and lowered slowly, the spring stretches a distance d. The mass is now displaced from its equilibrium position and undergoes 100 oscillations in 48.9 s. What is the stretch distance d?

Accepted Answer

The answer of When a mass of 0.350 kg is...

Question 17

An 85.0-kg man climbs onto a 900-kg car with worn out shock absorbers and whose springs have an effective force constant of 125 kN/m. As he drives along, he hits a bump, and this starts the car oscillating with simple harmonic motion. What is the smallest amplitude of the oscillations that will cause him to bounce from his seat?

Accepted Answer

The smallest amplitude that will cause the man to bounce from his seat is when the acceleration of the car equals the acceleration due to gravity. Using the formula for maximum acceleration in simple harmonic motion, $ a_{max} = \omega^2 A $, where $ \omega = \sqrt{\frac{k}{m}} $, and setting $ a_{max} = g $, we solve for $ A $. With $ k = 125,000 \, \text{N/m} $ and $ m = 985 \, \text{kg} $, we find $ A \approx 7.73 \, \text{cm} $.

Question 18

The position of an air-track cart that is oscillating on a spring is given by x = (12.4 cm) cos[(6.35 s^-1)t]. At what value of t after t = 0 s is the cart first located at x = 8.47 cm?

Accepted Answer

Given equation: x = (12.4 cm) cos[(6.35 s^-1)t]

We are looking for the time at which x = 8.47 cm.

So we need to solve:

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

cos[(6.35 s^-1)t] = 8.47/12.4 = 0.6831

Using inverse cosine function, we get:

(6.35 s^-1)t = cos^-1(0.6831)

t = cos^-1(0.6831) / 6.35 s^-1

t = 0.1288 s

Therefore, the cart will first be located at x = 8.47 cm after t = 0.1288 s.

The closest answer choice is C) 0.129 s.

Question 19

A simple harmonic oscillator has an amplitude of 3.50 cm and a maximum speed of 26.0 cm/s. What is its speed when the displacement is 1.75 cm?

Accepted Answer

The answer of A simple harmonic oscillator has an amplitude...

Question 20

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 is the maximum speed of the needle?

Accepted Answer

The equation for the velocity of a mass undergoing simple harmonic motion is:

v = Aωcos(ωt)

Where A is the amplitude, ω is the angular frequency (ω = 2πf), and t is time.

At the maximum displacement (i.e. at the amplitude), the velocity is zero. Therefore, the maximum velocity occurs at the equilibrium position (where the displacement is zero).

Thus, when t = 0, we have:

v = Aωcos(0) = Aω

Plugging in the values given, we get:

v = (1.27 cm)(2π(2.55 Hz)) ≈ 20.3 cm/s

Therefore, the maximum speed of the needle is 20.3 cm/s, which corresponds to answer choice C.

Quiz 13: Oscillations About Equilibrium

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46 s^-1) t]. What is the angular frequency for this motion?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. In any 4.60 s time interval, how much time is the mass more than 4.15 cm away from x = 0 cm?

When a mass of 0.350 kg is attached to a vertical spring and lowered slowly, the spring stretches 12.0 cm. The mass is now displaced from its equilibrium position and undergoes simple harmonic oscillations. What is the period of the oscillations?

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 is the maximum acceleration of the needle?

A 92-kg man climbs onto a car with worn out shock absorbers and this causes the car to drop down 4.5 cm. As he drives along, he hits a bump, and this starts the car oscillating at an angular frequency of 4.52 rad/s. What is the mass of the car?

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. How long does it take it to travel 11.43 cm?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is its speed at t = 2.50 s?

A mass of 0.150 kg is attached to a spring with a force constant of 3.58 N/m and undergoes simple harmonic oscillations. What is the period of the oscillations?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s²) t]. What is the acceleration of the mass when t = 0.825 s?

In a supermarket, you place a 5 lb (22.3 N) bag of oranges on a scale. The scale starts to oscillate at 2.70 Hz. What is the force constant of the scale?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is the value of t the first time after t = 0 s that the mass is at x = 4.15 cm?

An air-track cart is attached to a spring and completes one oscillation every 5.67 s. At t = 0 s the cart is released at a distance of 0.250 m from its equilibrium position. What is the position of the cart when t = 29.6 s?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is its acceleration at t = 2.50 s?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s^-1) t]. What is the speed of the mass when t = 0.805 s?

A mass of 0.250 kg is attached to a spring and undergoes simple harmonic oscillations with a period of 0.640 s. What is the force constant of the spring?

When a mass of 0.350 kg is attached to a vertical spring and lowered slowly, the spring stretches a distance d. The mass is now displaced from its equilibrium position and undergoes 100 oscillations in 48.9 s. What is the stretch distance d?

The position of an air-track cart that is oscillating on a spring is given by x = (12.4 cm) cos[(6.35 s^-1) t]. At what value of t after t = 0 s is the cart first located at x = 8.47 cm?

A simple harmonic oscillator has an amplitude of 3.50 cm and a maximum speed of 26.0 cm/s. What is its speed when the displacement is 1.75 cm?

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 is the maximum speed of the needle?

Quiz 13: Oscillations About Equilibrium

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46 s-1) t]. What is the angular frequency for this motion?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. In any 4.60 s time interval, how much time is the mass more than 4.15 cm away from x = 0 cm?

When a mass of 0.350 kg is attached to a vertical spring and lowered slowly, the spring stretches 12.0 cm. The mass is now displaced from its equilibrium position and undergoes simple harmonic oscillations. What is the period of the oscillations?

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 is the maximum acceleration of the needle?

A 92-kg man climbs onto a car with worn out shock absorbers and this causes the car to drop down 4.5 cm. As he drives along, he hits a bump, and this starts the car oscillating at an angular frequency of 4.52 rad/s. What is the mass of the car?

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. How long does it take it to travel 11.43 cm?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is its speed at t = 2.50 s?

A mass of 0.150 kg is attached to a spring with a force constant of 3.58 N/m and undergoes simple harmonic oscillations. What is the period of the oscillations?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s2) t]. What is the acceleration of the mass when t = 0.825 s?

In a supermarket, you place a 5 lb (22.3 N) bag of oranges on a scale. The scale starts to oscillate at 2.70 Hz. What is the force constant of the scale?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is the value of t the first time after t = 0 s that the mass is at x = 4.15 cm?

An air-track cart is attached to a spring and completes one oscillation every 5.67 s. At t = 0 s the cart is released at a distance of 0.250 m from its equilibrium position. What is the position of the cart when t = 29.6 s?

A mass is oscillating on a spring with a period of 4.60 s. At t = 0 s the mass has zero speed and is at x = 8.30 cm. What is its acceleration at t = 2.50 s?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s-1) t]. What is the speed of the mass when t = 0.805 s?

A mass of 0.250 kg is attached to a spring and undergoes simple harmonic oscillations with a period of 0.640 s. What is the force constant of the spring?

When a mass of 0.350 kg is attached to a vertical spring and lowered slowly, the spring stretches a distance d. The mass is now displaced from its equilibrium position and undergoes 100 oscillations in 48.9 s. What is the stretch distance d?

The position of an air-track cart that is oscillating on a spring is given by x = (12.4 cm) cos[(6.35 s-1) t]. At what value of t after t = 0 s is the cart first located at x = 8.47 cm?

A simple harmonic oscillator has an amplitude of 3.50 cm and a maximum speed of 26.0 cm/s. What is its speed when the displacement is 1.75 cm?

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 is the maximum speed of the needle?

The position of a mass that is oscillating on a spring is given by x = (17.4 cm) cos[(5.46 s^-1) t]. What is the angular frequency for this motion?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s²) t]. What is the acceleration of the mass when t = 0.825 s?

The position of a mass that is oscillating on a spring is given by x = (12.3 cm) cos[(1.26 s^-1) t]. What is the speed of the mass when t = 0.805 s?

The position of an air-track cart that is oscillating on a spring is given by x = (12.4 cm) cos[(6.35 s^-1) t]. At what value of t after t = 0 s is the cart first located at x = 8.47 cm?