Question

If a magnet is attached to a string R long is pulled out to one side so the string is horizontal . When the magnet is let go , It swings downward and strikes an iron cube B of the same mass that is resting on a frictionless surface . The two sticks together and swing upward on the other side . What is the maxium value of theta , the angle between the string and the vertical ?

Answer 1

According to the law of conservation of energy, the sum of the potential energy and kinetic energy of a pendulum is conserved. Hence, the potential energy at the maximal angle of deflection (when the kinetic energy is zero) is equal to the kinetic energy at the downward position (when the potential energy is taken to be zero). Let $v$ be the velocity of the pendulum body at the downward position, and let $h$ be its height at the maximal angle of deflection. Then the law of conservation of energy is expressed as $m v^2/2 = m g h$, whence $h = v^2/2 g$. As the magnet sticks to the iron cube, the mass of the newly formed pendulum body increases twice. Since the cube was initially at rest, the velocity of the formed body at this instant is two times smaller than the velocity of the magnet before the collision. By the previous equation, the newly formed pendulum body will rise to 1/4 of the maximum height of the original pendulum, which was equal to $R$. The angle $\theta$ between the string and the vertical in this position is then determined from the equation

$$h = R (1 - \cos \theta) = \frac{R}{4} \, ,$$

or $\cos \theta = 3/4$. Hence $\theta = \arccos (3/4) \approx 41.4^\circ$.

Answer: $41.4^\circ$.