AP Physics 1 : AP Physics 1

Study concepts, example questions & explanations for AP Physics 1

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Example Questions

Example Question #5 : Angular Momentum

A comet is orbiting a star in a far distant galaxy wth only gravity acting upon it. After a period of time, the sun's radiation partially melts the comet, decreasing the mass to a fourth of what it was. Simultaneously, because of an irregular orbit, its distance to the star doubles. What expression best describes the comet's angular velocity, , compared to its original conditions (before the melting/distance change)?

Possible Answers:

Correct answer:

Explanation:

Lets examine the equation , where  = angular momentum,  = mass of the satellite,  = its angular velocity, and  = its radius from the center of its orbit. Because of a lack of outside forces (besides the gravity of its star) acting on the satellite, we know that its angular momentum is conserved. Therefore, as its mass is divided by four, and its radius doubled, its angular velocity must also double, making the correct answer .

Example Question #2 : Angular Momentum

Consider a  disk  rotating with an angular speed of . At a later time, a  ring  is suddenly placed on top of the disk. Calculate the new angular speed of the system.

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we will be using the conservation of angular momentum.  Initially we only have a disk rotating.  This gives us information about the initial angular momentum of the system.  Later, we have both a disk and a ring rotating together at some new angular velocity.  

Example Question #73 : Circular, Rotational, And Harmonic Motion

The moment of inertia of two masses separated by distance  is

Two balls, each of mass are attached by a massless rod of length  and spinning at  around the point directly on the midpoint of the rod. Then rod then stretches to a length of . Assuming no loss of momentum and that the axis remains constant, determine the new rotational velocity.

Possible Answers:

Correct answer:

Explanation:

First convert  into :

Conservation of angular momentum:

Where, in this case:

Combine equations:

Plug in values and solve for the final rotational velocity:

Example Question #3 : Angular Momentum

Consider the following system:

 

Spinning rod with masses at end

Two spherical masses, A and B, are attached to the end of a rigid rod with length l. The rod is attached to a fixed point, p, which is at the midpoint between the masses and is at a height, h, above the ground. The rod spins around the fixed point in a vertical circle that is traced in grey.  is the angle at which the rod makes with the horizontal at any given time ( in the figure).

The rod is spinning, and as it goes through its horizontal position, the total angular momentum, including both masses, is . What is the instantaneous linear velocity of mass A?

Neglect the mass of the rod.

Possible Answers:

Correct answer:

Explanation:

Since we are given the angular momentum, let's being with that and its formula:

Where:

L = angular momentum

I = moment of inertia

w = angular velocity

Since there are two masses on the rod (and we are neglecting the mass of the rod itself), we can expand this expression to:

Since the masses are attached to a rigid rod and going in a uniform circle, we can say that: 

 Since we're asked to determine the velocity of mass A, we'll substitute the angular velocity of mass A for mass B:

Rearranging for the angular velocity of A:

Now we need to determine what the moment of inertia is for each mass. Since they are spheres, we can treat them as point masses and use the following formula:

Thus for each mass, we get:

Where r is half the length of the rod and equal for both masses. Plugging these into the equation, we get:

Substituting half the length of the rod in for radius, we get:

We know the value of each variable in this equation, so we can solve it. But first, let's check to make sure our units are correct. The units for angular velocity are :

It checks out, so time to plug and chug:

We can then use the expression for angular velocity to get linear velocity:

Example Question #4 : Angular Momentum

Consider the following system:

Spinning rod with masses at end

Two spherical masses, A and B, are attached to the end of a rigid rod with length l. The rod is attached to a fixed point, p, which is at a height, h, above the ground. The rod spins around the fixed point in a vertical circle that is traced in grey.  is the angle at which the L side of the rod makes with the horizontal at any given time ( in the figure and can be negative if mass A is above the horizontal).

The system is currently at rest and being held so that mass A is below the horizontal and . The rod is then released and allowed to rotate freely. What is the maximum angular momentum achieved by mass B? Neglect air resistance and internal frictional forces

Possible Answers:

Correct answer:

Explanation:

We can begin with the expression for conservation of energy to solve this problem:

Since we are told that the system is initially at rest, we can eliminate initial kinetic energy to get expression (1):

Expanding each term from left to right to:

We are told that mass A is held at an angle of 15 degrees below the horizontal, so we can use the sine function to determine the height of each mass. We will assume that the lowest point of rotation has a height of 0.

Where d is the vertical distance below horizontal, and r is the radius from the center of the rod, which is a distance equal to half the length of the rod:

Rearranging for d:

We know that this is the distance below horizontal that mass A begins at and the distance above the horizontal that mass B begins at. Also, horizontal is at a height of half a length rod above our reference height, so we can say:

Plugging these into our expanded expression for initial potential energy, we get:

Moving on to final potential energy:

We are asked to find the maximum angular momentum achieved by mass B. This occurs when mass B is traveling at it's highest velocity, which is when the larger of 2 masses, mass A, is traveling through the low point of rotation. This means that mass A is at a height of 0, and thus has no final potential energy. We can then say:

Also, we know that mass B will be at the high point of rotation, which means that its height will be the length of the rod. Therefore, we can say:

Moving on to final kinetic energy:

We don't have to expand the velocity term since the two masses are always traveling at the same speed.

Now, substituting our expanded terms back into expression (1), we get:

In order to determine angular momentum, we are going to need to solve this equation for final velocity, so let's begin rearranging:

We have all of these values, so time to plug and chug:

This is the maximum velocity that mass B will experience. We can now use the expression for angular momentum:

where:

Where r is half the length of the rod:

We have all of these values, so time to plug and chug:

Example Question #31 : Circular And Rotational Motion

Two cars are racing side by side on a perfectly circular race track. The inner car is from the center of the track. The outer car is from center of the track.

If the outer car is moving at , determine it's angular momentum.

Possible Answers:

Correct answer:

Explanation:

Example Question #32 : Circular And Rotational Motion

Moment of inertia of disk:

A car with wheels of mass and wheels of radius is traveling at . Treating the wheels as disks of uniform mass density, calculate the angular momentum of one wheel.

Possible Answers:

None of these

Correct answer:

Explanation:

Finding angular velocity:

Using

and

Where is angular momentum

Combining equations and plugging in values:

Example Question #981 : Newtonian Mechanics

A solid bowling ball of mass  and radius  is traveling at a linear velocity of . What is the angular momentum of the ball?

Possible Answers:

Correct answer:

Explanation:

The expression for angular momentum is:

Where I is the moment of inertia for a solid sphere:

And w is the angular velocity:

Plugging these in, we get:

Plugging in our values, we get:

Example Question #11 : Angular Momentum

A bowling ball of mass  and radius  is traveling down a slope with a vertical height change of . If the ball starts from rest, what is the final linear velocity of the ball as it reaches the bottom of the slope? Neglect air resistance.

Possible Answers:

Correct answer:

Explanation:

Let's begin with the expression for conservation of energy:

Since the ball begins from rest, we can eliminate initial kinetic energy. Also, if we assume that the bottom of the slope has a height of 0, we can eliminate final potential energy to get:

Plugging in expressions, we get:

Where I is the moment of inertia for a solid sphere:

And w is the angular velocity:

Plugging these in, we get:

Rearranging for final velocity, we get:

Plugging in our values, we get:

Example Question #31 : Circular And Rotational Motion

Two identical cars are racing side by side on a circular race track. Which has the greater angular momentum?

Possible Answers:

The inside car

They are the same

The outside car

Impossible to determine

Correct answer:

The outside car

Explanation:

Where  is the radius of the circle

 is the mass of the object 

 is the linear velocity of the object

The car on the outside has a larger  and a larger , and the same , thus it has a higher angular momentum.

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