# AP Physics 1 : Other Harmonic Systems

## Example Questions

### Example Question #1 : Other Harmonic Systems

You push on a door with a force of 1.3 N at an angle of 45 degrees to the surface and at a distance of 0.5 m from the hinges. What is the torque produced?

Explanation:

To calculate torque, this equation is needed:

Next, identify the given information:

Plug these numbers into the equation to determine the torque:

### Example Question #2 : Other Harmonic Systems

If a mass on the end of a string of length 8 cm is pulled 30 degrees away from vertical, what will its speed be when the string is vertically alligned if the mass is released from rest?

Explanation:

We can determine the vertical components by using some geometry.

Since this problem boils down to being a problem of conservation of energy, we can state that

Since the mass is released from rest and we can state at the bottom of its arc it will be at height 0 m, this equation can be simplified:

is equal to the height of the mass above 0 m; therefore, by referring to the diagram, we can determine that . By plugging this in and rearranging the equation, we can solve for the speed of the mass when the string is vertically aligned:

### Example Question #1 : Other Harmonic Systems

A ball with mass 5 kg is attached to a spring and is released 10 meters from equilibrium. After some time, the ball passes the equilibrium point moving at . What is the spring constant () of the spring?

Explanation:

When the object is released, it has no kinetic energy (it isn't moving) and a potential energy of

When the object passes through the equilibrium point, it has no potential energy () and a kinetic energy of

Due to the conservation of energy, these two quantities must be equal to each other:

### Example Question #1 : Other Harmonic Systems

A 30 kg weight attached to a spring is at equilibrium lying horizontally on a table. The spring is lifted up and is stretched by 80 cm before the weight is lifted off the table. What is this spring's spring constant ()?

Explanation:

First, we should convert 80 centimeters to 0.8 meters.

We know the force applied to the weight by gravity is

The force applied by the spring in the opposite direction must be equal to this:

### Example Question #5 : Other Harmonic Systems

Instead of using a second hand to count seconds, a watchmaker decides to construct a simple harmonic system involving a mass and a spring suspended from a table.

Ignoring the effects of gravity, if the spring the watchmaker selects has a spring constant of , how large of a mass in kilograms should he attach to the spring such that the harmonic system does not oscillate too slow or too fast?

Explanation:

The angular velocity of the harmonic system is equal to the square root of the spring constant over the mass:

Since we need the angular velocity to be  and we are given the spring constant as , then we can set up an equation:

### Example Question #6 : Other Harmonic Systems

The position of a  mass in an oscillating spring-mass system is given by the following equation:

, where  is measured in , and  is measured in .

What is the velocity equation of the system as a function of time?