Question

Imagine two circular plates; one is solid and the other has a hole cut out of the center. Both plates have the same radius, same thickness, and same mass. The same force F is applied tangential to the edge of each plate in such a way that the plates rotate about an axis passing through the center and perpendicular to the surface of the plates.
Which one of the following statements is true regarding the angular acceleration?
1. Both plates will rotate with the same angular acceleration.
2. The solid plate will have the greater angular acceleration.
3. The plate with the hole will have the greater angular acceleration.
Which of the following statements helps to explain the question asked above? (Select all that apply.)
1. Because both plates have the same mass, they will have the same moment of inertia.
2. Angular acceleration is inversely proportional to the moment of inertia.
3. Angular acceleration is directly proportional to the moment of inertia.
4. The plate with the hole has its mass distributed further out from the axis of rotation, which will increase its moment of inertia.
5. Both plates will be subjected to the same torque.

Answer 1

Answer:

the correct statement is 2. The solid plate will have the greater angular acceleration.

the correct phrase is 4. The plate with the hole has its mass distributed further out from the axis of rotation, which will increase its moment of inertia.

Explanation:

Newton's second law expression for rotational motion is

         τ = I α         (1)

where the torque is

         τ = F r

in this case, as the discs have the same radius and the applied force is the same, the torque is the same on the two discs.

The moment of inertia is given by the expression

        I =∫ r² dm

for bodies with high symmetry are tabulated

the moment of inertia for in disk solid is     I₁ = ½ m R₂²

the moment for a disk with a hole               I₂ = ½ m (R₁² + R₂²)

We can see that the moment of inertia of the disk with the hole is greater than the moment of inertia of the solid disk.

Let's use equation 1

          α = τ/I

therefore the angular acceleration is lower for the body with the higher moment of inertia, consequently the solid disk has higher angular acceleration

the correct statement is 2

The reason is because the moment of inertia is higher for the hollow disk.

the correct phrase is 4