Question

A dart and a rubber sphere are suspended from the same point by light strings. When the dart is released from the position shown above, it swings downward until its tip strikes the sphere. The dart can have different tips attached. Tip A will stick into the rubber sphere upon impact. Tip B will bounce inelastically off the sphere. Tip C will bounce elastically off the sphere. For which tip, if any, will the sphere swing to the greatest maximum angle?

1) Tip A
2) Tip B
3) Tip C
4) The sphere will not swing to any maximum angle

Answer 1

A dart and a rubber sphere are suspended from the same point by light strings. When the dart is released from the position shown above, it swings downward until its tip strikes the sphere. The dart can have different tips attached. Tip A will stick into the rubber sphere upon impact. Tip B will bounce inelastically off the sphere. Tip C will bounce elastically off the sphere. For 3) Tip C , if any, will the sphere swing to the greatest maximum angle

Step-by-step explanation:

When the dart strikes the rubber sphere, the conservation of angular momentum plays a crucial role in determining the maximum angle to which the sphere swings.

For Tip A, where the dart sticks into the sphere, some angular momentum is transferred to the sphere, causing it to swing less compared to the other scenarios. For Tip B, where the dart bounces inelastically off the sphere, some energy is lost during the collision, reducing the overall angular momentum.

However, for Tip C, where the dart bounces elastically off the sphere, both linear and angular momentum are conserved, resulting in the sphere swinging to the greatest maximum angle.

Now, let's delve into the physics behind this. Angular momentum is given by the equation
`\(L = I\omega\),` where
`\(L\)` is angular momentum,
`\(I\)` is the moment of inertia, and
`\(\omega\)` is angular velocity. In an elastic collision, both linear and angular momentum are conserved. The rubber sphere's moment of inertia is affected less when the dart bounces elastically (Tip C), leading to the greatest swing.

In conclusion, Tip C, where the dart bounces elastically off the sphere, will result in the rubber sphere swinging to the greatest maximum angle due to the conservation of angular momentum during an elastic collision.