Two objects of equal mass are on a turning wheel. Mass 1 is located at the rim of the wheel while mass 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.
The tangential acceleration of mass 1 is______________the tangential acceleration of mass 2. (equal to?)
The centripetal (radial) acceleration of mass 1 is______________the centripetal acceleration of mass 2. (greater than?)
For a given time, the angle covered by mass 2 is______________the angle covered by mass 1. (equal to?)
The magnitude of the total acceleration of mass 1 is_____________the total acceleration of mass 2. (greater than?)
The moment of inertia of mass 1 about the axis of rotation is_____________the moment of inertia of mass 2 about the axis of rotation . (greater than?)
For a given time, mass 1 travels a distance that is____________the distance traveled by mass 2. (greater than?)
The speed of mass 1 is____________the speed of mass 2. (greater than?)
The angular acceleration of mass 2 is________________the angular acceleration of mass 1. (equal to?)
To answer these questions, we need to understand the concepts of tangential acceleration, centripetal acceleration, moment of inertia, and angular acceleration.
1. The tangential acceleration of an object is the rate at which its tangential velocity changes. Since both objects are on the same turning wheel, they have the same angular acceleration. Therefore, the tangential acceleration of mass 1 is equal to the tangential acceleration of mass 2.
2. The centripetal acceleration of an object is the acceleration towards the center of the circular path it follows. In this case, since mass 1 is located at the rim of the wheel, it is farther from the axis of rotation compared to mass 2. Due to this increased distance, the centripetal acceleration of mass 1 is greater than the centripetal acceleration of mass 2.
3. The angle covered by an object is directly proportional to its angular displacement. Since both masses are attached to the same wheel and are rotating together, they will cover the same angle in the same amount of time. Therefore, the angle covered by mass 2 is equal to the angle covered by mass 1.
4. The total acceleration of an object is the vector sum of its tangential acceleration and centripetal acceleration. Since the tangential acceleration of mass 1 is equal to the tangential acceleration of mass 2 (as explained in question 1), and the centripetal acceleration of mass 1 is greater than the centripetal acceleration of mass 2 (as explained in question 2), the magnitude of the total acceleration of mass 1 is greater than the total acceleration of mass 2.
5. The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass and distribution of mass around the axis of rotation. In this case, since mass 1 is located at the rim of the wheel and has a greater distance from the axis of rotation compared to mass 2, the moment of inertia of mass 1 is greater than the moment of inertia of mass 2.
6. Since both masses are rotating together on the same wheel, they will cover the same angle in the same amount of time. Therefore, the distance traveled by mass 1 is equal to the distance traveled by mass 2.
7. The speed of an object is the magnitude of its velocity. Since both masses are rotating together on the same wheel, their angular velocities will be the same. As mass 1 is located at the rim of the wheel, it has a greater linear distance to cover in the same amount of time compared to mass 2. Therefore, the speed of mass 1 is greater than the speed of mass 2.
8. The angular acceleration of an object is the rate at which its angular velocity changes. Since both objects are on the same turning wheel, they have the same angular acceleration. Therefore, the angular acceleration of mass 2 is equal to the angular acceleration of mass 1.