Question

A clay sphere is traveling horizontally with speed when it strikes a rod of length that is suspended from one end and has a rotational inertia of about its pivot point, as shown above. The clay sphere sticks to the rod, which begins to swing about the pivot. If both objects have mass , the initial angular speed of the sphere-rod system just after the collision is?

1) the fraction v over 2 l
2) the fraction 3 v over 4 l
3) the fraction 3 v over 2 l
4) the fraction 2 v over l

Answer 1

To find the initial angular velocity of the sphere-rod system after the collision, we can use the principle of conservation of angular momentum.

Step-by-step explanation:

To solve this problem, we can use the principle of conservation of angular momentum. Before the collision, the clay sphere has a linear momentum given by its mass (m) and speed (v). The moment of inertia (I) of the rod is also given. When the clay sphere sticks to the rod, the total angular momentum of the system is conserved. We can calculate the initial angular velocity of the sphere-rod system just after the collision using the formula:

Initial angular velocity = (m * v) / (I + m * l^2)

Where l is the length of the rod. Substitute the given values to find the correct answer.

Answer 2

The initial angular velocity of the sphere-rod system after the collision is the fraction of v over 4 l.

Step-by-step explanation:

When the clay sphere collides and sticks to the rod, the momentum of the system is conserved. Initially, the sphere has a linear momentum of mv, where m is its mass and v is its speed. After the collision, the system starts rotating about the pivot point. The angular momentum of the system is given by Iω, where I is the rotational inertia of the rod and ω is the angular velocity of the system. Since the sphere sticks to the rod, they both have the same angular velocity.

In this case, the initial angular velocity of the sphere-rod system just after the collision can be determined using the principle of conservation of angular momentum. We know that the linear momentum of the sphere before the collision is m v. The linear momentum is equal to the product of the mass and the velocity. After the collision, this linear momentum is converted into angular momentum, which is equal to the product of the moment of inertia I and the angular velocity ω. Therefore, we can set up the equation:

m v = I ω

Substituting the given values into the equation and solving for ω:

m v = 4 m l^2 ω

ω = v / 4 l^2

So, the initial angular velocity of the sphere-rod system just after the collision is the fraction of v over 4 l.