Question

A footballer jumps into the air with his back facing the goal and kicks a ball mid-air by twisting his body. At the moment of the kick:

• His center of mass is moving horizontally with velocity $v_{cm}$ = 4, m/s.

• He rotates his torso about a vertical axis at angular speed $\omega$ = 5, rad/s.

• His moment of inertia about the axis through the center of mass is $I$ = 2, kg $m^2$.

• He stops rotating right after the kick.

• Assume the foot applies an impulse $J$ = 20, Ns to the ball in the direction of the goal.

Q1. What is the angular momentum of the player just before the kick?

Q2. Assuming conservation of angular momentum, what torque was involved?

Answer 1

Q1. The angular momentum of the player just before the kick is 10 kg m$^2$/s (or 10 Nms). Q2. Assuming conservation of angular momentum, the net external torque involved is 0 Nm.

Answer 2

• Q1. What is the angular momentum of the player just before the kick?

The angular momentum ($L$) of a rotating body is given by the product of its moment of inertia ($I$) and its angular velocity ($\omega$). Given: Moment of inertia, $I = 2 \text{ kg m}^2$ Angular speed, $\omega = 5 \text{ rad/s}$

The angular momentum of the player just before the kick is: $L = I \omega$ $L = (2 \text{ kg m}^2) \times (5 \text{ rad/s})$ $L = 10 \text{ kg m}^2/\text{s}$

• Q2. Assuming conservation of angular momentum, what torque was involved?

The principle of conservation of angular momentum states that the total angular momentum of a system remains constant if the net external torque acting on the system is zero. In this problem, if we consider the system to be the footballer and the ball, the kick involves internal forces and torques between the footballer's foot and the ball. These internal torques cause a transfer of angular momentum from the footballer (who stops rotating) to the ball (which gains angular momentum). However, internal torques do not change the total angular momentum of the entire system. Since the question explicitly states "Assuming conservation of angular momentum", it implies that the net external torque acting on the combined system (footballer + ball) is zero.

Therefore, the net external torque involved is 0 Nm.

Solution:

Q1. What is the angular momentum of the player just before the kick? The angular momentum $L$ is given by the formula: $L = I \omega$ Given $I = 2 \text{ kg m}^2$ and $\omega = 5 \text{ rad/s}$. $L = (2 \text{ kg m}^2) \times (5 \text{ rad/s}) = 10 \text{ kg m}^2/\text{s}$

Q2. Assuming conservation of angular momentum, what torque was involved? According to the principle of conservation of angular momentum, if the total angular momentum of a system is conserved, then the net external torque acting on that system must be zero. The interaction between the player and the ball is an internal one within the player-ball system. Internal torques do not affect the total angular momentum of the system. Therefore, the net external torque on the system is zero. $\tau_{net, ext} = 0 \text{ Nm}$