What happens to centripetal acceleration when speed is doubled and angular velocity is halved

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Troy spins a ball attached to a string around in a vertical circle. Assume the ball travels at constant speed during its circular path. Which of the following is not constant throughout the ball's path?

Possible Answers:

The rate at which the string does work on the ball

Force of gravity on the ball

Angular momentum of the ball

All of these are constant

Correct answer:

Tension in the string

Explanation:

Let's go through each answer choice one by one. The force of gravity on the ball is always its mass times gravity, which is constant. The string's pull on the ball is always perpendicular to the ball's motion, so it isn't doing any work, and thus has a constant rate of work

. The ball's angular momentum is always constant since its mass, the radius of the circle it's traveling in, and its speed are constant. The only the changing is the tension in the string. This is true because the ball needs the same center pulling force (centripetal force) throughout its path, but this centripetal force is constantly changing direction. For example, at the top of its path, the centripetal force must be down because that's the direction of the center of its circle of motion, but at the bottom, the centripetal force must be up. There are two forces on the ball, gravity and tension. The force of gravity is always down and constant, so the tension needs to change in order to keep the center pulling force the same. For example, when the ball is at the bottom of the circle, the force of gravity is away from the center of the circle, so tension must be higher to keep the center pull constant.

An airplane is traveling at

and performs a vertical loop in

. Determine the net force on the

pilot at the bottom of the loop.

Possible Answers:

Correct answer:

Explanation:

Circumference of a circle:

Distance plane travels during loop (which will be the circumference)

Centripetal acceleration:

Plugging in values:

An airplane is traveling at

and performs a vertical loop in

. Determine the force due to the seat on the

pilot at the bottom of the loop

Possible Answers:

Correct answer:

Explanation:

Circumference of a circle:

Distance plane travels during loop (which will be the circumference)

Centripetal acceleration:

Plugging in values:

An airplane is traveling at

and performs a vertical loop in

. Determine the radius of the loop.

Possible Answers:

Correct answer:

Explanation:

Circumference of a circle:

Distance plane travels during loop (which will be the circumference)

model train completes a circle of radius

. Determine the centripetal acceleration.

Possible Answers:

Correct answer:

Explanation:

One circle is equal to

, thus the linear velocity is:

Using

model train completes a circle of radius

. Determine the centripetal force.

Possible Answers:

Correct answer:

Explanation:

One circle is equal to

, thus the linear velocity is

Using

Use Newton's second law:

model train completes a circle of radius

. Determine the angular momentum.

Possible Answers:

Correct answer:

Explanation:

One circle is equal to

, thus the linear velocity is:

The definition of angular momentum:

Plugging in values:

An object has a mass M, velocity V, and moves in a circle with radius R.

What happens to the centripetal acceleration on the object when the mass is doubled?

Possible Answers:

The centripetal acceleration is quadrupled.

The centripetal acceleration is halved.

The centripetal acceleration is doubled.

The centripetal acceleration remains the same.

Correct answer:

The centripetal acceleration remains the same.

Explanation:

The equation for centripetal acceleration is:

Where

is the centripetal acceleration on an object,

is the velocity of an object, and

is the radius in which the object moves in a circle.

We can see that mass does not play a role in the centripetal acceleration of an object, so no matter what happens to the mass, the centripetal acceleration remains the same.

An object has a mass M, velocity V, and moves in a circle with radius R.

What happens to the centripetal acceleration on the object when the radius is halved?

Possible Answers:

The centripetal acceleration of the object goes to zero.

The centripetal acceleration of the object is halved.

The centripetal acceleration of the object is quadrupled.

The centripetal acceleration of the object remains the same.

The centripetal acceleration of the object is doubled.

Correct answer:

The centripetal acceleration of the object is doubled.

Explanation:

The equation for centripetal acceleration is:

Where

is the centripetal acceleration on an object,

is the velocity of an object, and

is the radius in which the object moves in a circle.

The radius has an inverse relationship with centripetal acceleration, so when the radius is halved, the centripetal acceleration is doubled.

An object has a mass M, velocity V, and moves in a circle with radius R.

What happens to the centripetal acceleration on the object when the velocity is doubled?

Possible Answers:

The centripetal acceleration of the object is quadrupled.

The centripetal acceleration of the object is doubled.

The centripetal acceleration of the object is halved.

The centripetal acceleration of the object remains the same.

The centripetal acceleration of the object goes to zero.

Correct answer:

The centripetal acceleration of the object is quadrupled.

Explanation:

The equation for centripetal acceleration is:

Where

is the centripetal acceleration on an object,

is the velocity of an object, and

is the radius in which the object moves in a circle.

The velocity has an quadratic relationship with centripetal acceleration, so when the velocity is doubled, the centripetal acceleration is quadrupled.

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Michael
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