Question 1

A 0.28-kg block on a horizontal frictionless surface is attached to an ideal massless spring whose spring constant is 500 N/m. The block is pulled from its equilibrium position at x = 0.00 m to a displacement x = +0.080 m and is released from rest. The block then executes simple harmonic motion along the horizontal x-axis. When the displacement is x = -0.052 m, find the acceleration of the block.

Accepted Answer

The acceleration in simple harmonic motion is given by a = -kx/m. Here, k = 500 N/m, x = -0.052 m, and m = 0.28 kg. Substituting these values gives a = -(500)(-0.052)/0.28 = 92.857 m/s², which rounds to 92 m/s².

Question 2

An object weighing 44.1 N hangs from a vertical massless ideal spring. When set in vertical motion, the object obeys the equation y(t) = (6.20 cm) cos[(2.74 rad/s)t - 1.40].
(a) Find the time for this object to vibrate one complete cycle.
(b) What are the maximum speed and maximum acceleration of the object?
(c) What is the TOTAL distance the object moves through in one cycle?
(d) Find the maximum kinetic energy of the object.
(e) What is the spring constant of the spring?

Accepted Answer

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Question 3

A 0.50-kg object is attached to an ideal massless spring of spring constant 20 N/m along a horizontal, frictionless surface. The object oscillates in simple harmonic motion and has a speed of 1.5 m/s at the equilibrium position.
(a) What is the amplitude of vibration?
(b) At what location are the kinetic energy and the potential energy of the system the same?

Accepted Answer

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Question 4

A 2.25-kg object is attached to a horizontal an ideal massless spring on a frictionless table. What should be the spring constant of this spring so that the maximum acceleration of the object will be g when it oscillates with amplitude of 4.50 cm?

Accepted Answer

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Question 5

A 12.0-N object is oscillating in simple harmonic motion at the end of an ideal vertical spring. Its vertical position y as a function of time t is given by y(t) = 4.50 cm cos[(19.5 s^-1)t - π/8]. (a) What is the spring constant of the spring? (b) What is the maximum acceleration of the object? (c) What is the maximum speed that the object reaches? (d) How long does it take the object to go from its highest point to its lowest point?

Accepted Answer

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Question 6

The x component of the velocity of an object vibrating along the x-axis obeys the equation v_x(t) = (0.445 m/s) sin[(25.4 rad/s)t + 0.223]. (a) What is the amplitude of the motion of this object? (b) What is the maximum acceleration of the vibrating object?

Accepted Answer

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Question 7

An object of mass 6.8 kg is attached to an ideal massless spring of spring constant 1690 N/m. The object is Calculate the maximum speed the object reaches during its motion.

Accepted Answer

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Question 8

An object that weighs 2.450 N is attached to an ideal massless spring and undergoes simple harmonic oscillations with a period of 0.640 s. What is the spring constant of the spring?

Accepted Answer

The period of a spring-mass system is given by:
$T=2\pi\sqrt{\frac{m}{k}}$, where $m$ is the mass attached to the spring and $k$ is the spring constant.

Rearranging the equation, we get:

$k=\frac{4\pi^2m}{T^2}$

Substituting the given values, we get:

$k=\frac{4\pi^2(2.450	ext{ N})}{(0.640	ext{ s})^2}=24.1	ext{ N/m}$

Therefore, the answer is C.

Question 9

A 1.6-kg block on a horizontal frictionless surface is attached to an ideal massless spring whose spring constant is 190 N/m. The block is pulled from its equilibrium position at x = 0.00 m to a displacement x = +0.080 m and is released from rest. The block then executes simple harmonic motion along the horizontal x-axis. What is the velocity of the block at time t = 0.40 s?

Accepted Answer

The velocity of the block at time t = 0.40 s is 0.82 m/s. This is calculated using the formula for velocity in simple harmonic motion: v(t) = -Aωsin(ωt), where A is the amplitude (0.080 m), ω is the angular frequency (√(k/m) = √(190/1.6)), and t is the time (0.40 s).

Question 10

An object of mass 8.0 kg is attached to an ideal massless spring and allowed to hang in the Earth's gravitational field. The spring stretches 3.6 cm before it reaches its equilibrium position. If this system is allowed to oscillate, what will be its frequency?

Accepted Answer

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Question 11

A machine part is vibrating along the x-axis in simple harmonic motion with a period of 0.27 s and a range (from the maximum in one direction to the maximum in the other) of 3.0 cm. At time t = 0 it is at its central position and moving in the +x direction. What is its position when t = 55 s?

Accepted Answer

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Question 12

A 1.5-kg mass attached to an ideal massless spring with a spring constant of 20.0 N/m oscillates on a horizontal, frictionless track. At time t = 0.00 s, the mass is released from rest at x = 10.0 cm. (That is, the spring is stretched by 10.0 cm.)
(a) Find the frequency of the oscillations.
(b) Determine the maximum speed of the mass. At what point in the motion does the maximum speed occur?
(c) What is the maximum acceleration of the mass? At what point in the motion does the maximum acceleration occur?
(d) Determine the total energy of the oscillating system.
(e) Express the displacement x as a function of time t.

Accepted Answer

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Question 13

In the figure, a 0.24-kg ball is suspended from a very light string 9.79 m long and is pulled slightly to the left. As the ball swings without friction through the lowest part of its motion it encounters an ideal massless spring attached to the wall. The spring pushes against the ball and eventually the ball is returned to its original starting position. Find the time for one complete cycle of this motion if the spring constant of the spring is 21 N/m. (Assume that once the pendulum ball hits the spring there is no effect due to the vertical movement of the ball.)

Accepted Answer

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Question 14

A 2.00-kg object is attached to an ideal massless horizontal spring of spring constant 100.0 N/m and is at rest on a frictionless horizontal table. The spring is aligned along the x-axis and is fixed to a peg in the table. Suddenly this mass is struck by another 2.00-kg object traveling along the x-axis at 3.00 m/s, and the two masses stick together. What are the amplitude and period of the oscillations that result from this collision?

Accepted Answer

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Question 15

A 56 kg bungee jumper jumps off a bridge and undergoes simple harmonic motion. If the period of oscillation is 11.2 s, what is the spring constant of the bungee cord, assuming it has negligible mass compared to that of the jumper?

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Question 16

A 0.025-kg block on a horizontal frictionless surface is attached to an ideal massless spring whose spring constant is 150 N/m. The block is pulled from its equilibrium position at x = 0.00 m to a displacement x = +0.080 m and is released from rest. The block then executes simple harmonic motion along the horizontal x-axis. When the displacement is x = 0.024 m, what is the kinetic energy of the block?

Accepted Answer

The kinetic energy of the block is given by:$$K = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2A^2\sin^2(\omega t)$$where m is the mass of the block, v is its velocity, $\omega$ is the angular frequency, A is the amplitude of the motion, and t is the time.At the given displacement, the kinetic energy is:$$K = \frac{1}{2}(0.025 	ext{ kg})(150 	ext{ N/m})(0.080 	ext{ m})^2\sin^2(\omega t) = 0.44 	ext{ J}$$

Question 17

A 0.25 kg ideal harmonic oscillator has a total mechanical energy of 4.0 J. If the oscillation amplitude is 20.0 cm, what is the oscillation frequency?

Accepted Answer

The oscillation frequency is given by f = 1/2π√(k/m), where k is the spring constant and m is the mass. The total mechanical energy is given by E = 1/2kA^2, where A is the oscillation amplitude. Substituting these expressions into the first equation, we get f = 1/2π√(2E/mA^2). Plugging in the given values, we get f = 1/2π√(2(4.0 J)/(0.25 kg)(0.20 m)^2) = 4.5 Hz.

Question 18

A frictionless simple pendulum on Earth has a period of 1.75 s. On Planet X its period is 2.14 s. What is the acceleration due to gravity on Planet X?

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Question 19

In the figure, two masses, M = 16 kg and m = 12.8.0 kg, are connected to a very light rigid bar and are attached to an ideal massless spring of spring constant 100 N/m. The system is set into oscillation with an amplitude of 78 cm. At the instant when the acceleration is at its maximum, the 16-kg mass separates from the 12.8.0-kg mass, which then remains attached to the spring and continues to oscillate. What will be the amplitude of oscillation of the 12.8.0-kg mass?

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Question 20

A 2.0 kg block on a frictionless table is connected to two ideal massless springs with spring constants k₁ and k₂ whose opposite ends are fixed to walls, as shown in the figure. What is angular frequency of the oscillation if k₁ = 7.6 N/m and k₂ = 5.0 N/m?

Accepted Answer

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Quiz 13: Periodic Motion