Two objects of equal mass are on a turning wheel. Object 1 is located at the rim of the wheel while object 2 is located halfway between the rim and the axis of rotation. The wheel is rotating with a non-zero angular acceleration.
For each of the following statements select the correct option to complete the statement.

1. The centripetal (radial) acceleration of object 1 is the centripetal acceleration of object 2.
2. The magnitude of the total acceleration of object 1 is the total acceleration of object 2.
3. The tangential acceleration of object 1 is the tangential acceleration of object 2.
4. For a given time, object 2 travels a distance that is the distance traveled by object 1.
5. The speed of object 1 is the speed of object 2.
6. For a given time, the angle covered by object 2 is the angle covered by object 1.
7. The angular acceleration of object 1 is the angular acceleration of object 2.

a) equal to
b) less than
c) greater than

Question

26-02-2020
Physics

contestada

Two objects of equal mass are on a turning wheel. Object 1 is located at the rim of the wheel while object 2 is located halfway between the rim and the axis of rotation. The wheel is rotating with a non-zero angular acceleration.
For each of the following statements select the correct option to complete the statement.

1. The centripetal (radial) acceleration of object 1 is the centripetal acceleration of object 2.
2. The magnitude of the total acceleration of object 1 is the total acceleration of object 2.
3. The tangential acceleration of object 1 is the tangential acceleration of object 2.
4. For a given time, object 2 travels a distance that is the distance traveled by object 1.
5. The speed of object 1 is the speed of object 2.
6. For a given time, the angle covered by object 2 is the angle covered by object 1.
7. The angular acceleration of object 1 is the angular acceleration of object 2.

a) equal to
b) less than
c) greater than

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Answer:

1. The centripetal (radial) acceleration of object 1 is greater than the centripetal acceleration of object 2.

2. The magnitude of the total acceleration of object 1 is greater than the total acceleration of object 2.

3. The tangential acceleration of object 1 is greater than the tangential acceleration of object 2.

4. For a given time, object 2 travels a distance that is less than the distance traveled by object 1.

5. The speed of object 1 is greater than the speed of object 2.

6. For a given time, the angle covered by object 2 is equal to the angle covered by object 1.

7. The angular acceleration of object 1 is equal to the angular acceleration of object 2.

Explanation:

It is better to explain the statements in mixed order.

7. The angular acceleration of both objects are equal to that of the wheel, since they are not moving with respect to the wheel, they are moving with the wheel.

6. Similarly, since both objects are moving with the wheel, they cover the same angle as they turn.

3. The relationship between tangential acceleration and angular acceleration is

[tex]\vec{a}_t = R\vec{\alpha}[/tex]

where α denotes the angular acceleration. Therefore; since the angular accelerations are equal, greater the radius greater the tangential acceleration.

5. Since the objects are at rest with respect to the wheel, the angular speed of both objects are equal to that of the wheel. Therefore, the tangential speed of object 1 is also greater than object 2, according to the following relationship:

[tex]v = R\omega[/tex]

4. Let's assume that they both complete one complete cycle. In that case, object 1 traveled a distance of 2πR and object 2 traveled a distance of 2π(R/2) = πR. Therefore, object 2 traveled a distance less than object 1.

1. The centripetal acceleration is given by the following formula

[tex]a = \frac{v^2}{R}[/tex]

Since we know from the statement 5, that the speed of object 1 is greater than the speed of object 2, the centripetal acceleration of object 1 is greater than that of object 2.

2. The magnitude of total acceleration is the magnitude of the vector sum of tangential acceleration and centripetal acceleration. Since both terms are greater for object 1, the total acceleration of object 1 is greater than that of object 2.