# AP Physics C: Mechanics : Harmonic Motion

## Example Questions

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### Example Question #1 : Understanding Principles Of Harmonic Motion

A 5.0 kg mass oscillates once. The total distance it travels is 1.5 m and it takes 4.0 s to travel that distance. What is its frequency of oscillation?

Explanation:

Frequency is only based on the period of the oscillation; all the other given information is useless for this problem.  Using , we can calculate that the frequency is 0.25 Hz.

### Example Question #1 : Understanding Principles Of Harmonic Motion

A mass oscillates on a spring with period . If the mass is doubled, what is the new period of oscillation?

Explanation:

The formula for the period of oscillation is

.

When we double the mass, we get:

Because the new factor of 2 is under the square root sign, and also in the numerator, the new period will be increased by .

### Example Question #21 : Motion

A mass is attached to the ceiling by a spring and hangs vertically downward. The mass is stretched downward so its amplitude is  units from its equilibrium position. What is the distance the mass travels during one full oscillation?

Explanation:

When the mass is at point , it hasn't traveled at all. When it reaches the spring's equilibrium point, it has traveled a distance of . The mass then continues to a point that's equal to the initial distance traveled, but on the opposite side of the equilibrium point, so the total distance traveled so far is . The mass must then travel back to the starting point to complete the oscillation, so the total distance traveled is .

### Example Question #21 : Ap Physics C

A mass is attached to a spring, which is fixed to a wall. The mass is pulled away from the spring's equilibrium point and is then released. At what point does the mass experience its maximum kinetic energy?

Equal in all places

At the spring's equilibrium point

Where the mass started

Halfway through one full oscillation

The mass does not have kinetic energy

At the spring's equilibrium point

Explanation:

The formula for determining kinetic energy is

So, kinetic energy will be greatest when the mass is moving most quickly. The force of the spring on the mass increases the mass's velocity until the spring’s equilibrium point, where the force of the spring acts against the motion of the mass, slowing it down. The mass is moving fastest at the spring's equilibrium point, so that's where its kinetic energy is greatest.

### Example Question #1 : Harmonic Motion

A mass is attached to a spring with a spring constant of .  The mass is moved 2 m from the spring's equilibrium point. What is the total energy of the system?

Explanation:

The total energy of the system is

but due to the face that the mass currently has no velocity, the kinetic energy term goes to zero.

Plugging in the given values, we can solve for the total energy of the system:

### Example Question #1 : Harmonic Motion

A mass  is attached to a spring of force constant . The mass rests on a frictionless surface and oscillates horizontally, with oscillations of amplitude . What is the maximum velocity of this mass in terms of , , and ?

Explanation:

Relevant equations:

Write expressions for the initial kinetic and potential energies, if the spring is initially stretched to the maximum amplitude before being released.

Write expressions for the final kinetic and potential energies when the spring crosses the equilibrium point.

Use conservation of energy to equate the initial and final energy sums.

Solve the equation to isolate .

### Example Question #1 : Harmonic Motion

A ball is attached to a spring on a frictionless, horizontal plane. If the spring constant is  and the mass of the ball three kilograms, at what angular frequency will the system oscillate?

Explanation:

The units for angular frequency, , are radians per second.

We need to derive the equation for angular frequency using conservation of energy.

Rearrange to solve for the velocity:

The velocity is also the product of angular frequency and the distance of the oscillation:

Use this equation to derive the equation for angular frequency:

Finally, use our given mass and spring constant to solve:

### Example Question #1 : Using Spring Equations

A 1kg ball is attached to a massless spring on a frictionless, horizontal plane. If at its equilibrium position, the ball is moving at , how much total energy is in the system?

Explanation:

At the equilibrium position, the spring does not contribute any potential energy. It is neither stretched, nor compressed.

All of the energy in the system is kinetic energy, resulting from the given velocity:

### Example Question #31 : Mechanics Exam

A 500g ball is attached to a massless spring on a frictionless, horizontal plane. If at its equilibrium position, the ball is moving at , and the spring constant is , what is the maximum displacement of the ball from its equilibrium position?

Explanation:

To solve this question, we will need to use conservation of energy. With no displacement, the ball has a velocity of  and zero displacement. At its maximum displacement, the velocity will be zero and all the energy will be converted to spring potential energy.

Use our given values to solve:

### Example Question #1 : Harmonic Motion

A bungee-jumping company operates on a bridge 200 m above the ground. They use bungee cords that are 100 m when they are unstretched; these bungee cords have a spring constant of .  What could the mass of their largest customer be before hitting the ground becomes an issue?