Pendulum and block. A pendulum consisting of a sphere of mass \(m\) attached to the end of a massless rope of length \(L\) is initially lifted so that the rope is tight and horizontal, and released. At the bottom of its path, the sphere hits a block of mass \(M=2 m\) lying at rest on a frictionless table. The collision is elastic. (a) Find the velocities of the sphere and the block immediately after the collision. (b) How high does the pendulum swing after the collision?

The concept of conservation of energy is central to understanding the motion of the pendulum before the collision. Initially, the pendulum is lifted to a height equal to the length of the rope L, giving it potential energy. This potential energy is given by the formula \[ U = mgh = mgL \] where g is the acceleration due to gravity, and h is the height. Upon release, the pendulum swings down and this potential energy is fully converted into kinetic energy at the bottom of its path. The kinetic energy is given by \[ K = \frac{1}{2} mv^2 \]. By the conservation of energy principle, the initial potential energy equals the kinetic energy at the bottom, \[ mgL = \frac{1}{2} mv^2 \]. Solving for the velocity, we get \[ v = \sqrt{2gL} \].
This step shows how energy conversion helps us find the speed of the pendulum before the collision.

An elastic collision is one in which both kinetic energy and momentum are conserved. In this problem, the sphere from the pendulum collides with a stationary block of mass 2m. Two key equations arise from this concept:

• Conservation of momentum: \( mv = mv_s + 2mv_b \)
• Conservation of kinetic energy: \( \frac{1}{2} mv^2 = \frac{1}{2} mv_s^2 + mv_b^2 \)

By solving these two equations together, we can find the velocities of the sphere and the block immediately after the collision. Using substitution, we substitute \( v_s = v - 2v_b \) into the kinetic energy equation and solve to find that \( v_b = \frac{v}{3} \). Substituting back, we find \( v_s = \frac{\sqrt{2gL}}{3} \). These results confirm the velocities while abiding by the principles of an elastic collision.

The conservation of momentum principle states that the total momentum of a closed system remains constant if no external forces act on it. Mathematically, it can be expressed for our situation as \[ p_{initial} = p_{final} \]. Initially, only the sphere has momentum since the block is at rest. So the initial momentum is \[ m \cdot v \]. After the collision, the sphere and the block both are in motion. Thus, the total momentum is the sum of their momenta: \[ mv_s + 2mv_b \]. Setting the initial momentum equal to the final momentum gives the equation \[ mv = mv_s + 2mv_b \]. This equation helps to relate the velocities of both the sphere and the block right after the collision. It must be solved alongside the energy conservation equation to obtain the final velocities.

Kinematics focuses on describing the motion of objects without considering the forces causing the motion. In the context of our pendulum problem, we need to determine the height the pendulum reaches after the collision. We know the velocity of the sphere right after the collision \( v_s = \frac{\sqrt{2gL}}{3} \). Using the conservation of energy again, we convert this kinetic energy back to potential energy as the pendulum swings upward: \[ \frac{1}{2} m v_s^2 = mgh \]. Substituting \( v_s \) we get: \[ \frac{1}{2} m \left( \frac{\sqrt{2gL}}{3} \right)^2 = mgh \]. Simplifying this, we find the height \[ h = \frac{L}{9} \]. Hence, by understanding the sphere's motion using kinematic principles, we determine how high the pendulum rises after the collision, bringing our analysis full circle.