A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.)

Verified step by step guidance

1. Calculate the moment of inertia of the turntable before the parachutist lands. Use the formula for the moment of inertia of a disk, which is $I = \frac{1}{2} M R^2$, where $M$ is the mass of the disk and $R$ is its radius.
2. Calculate the moment of inertia of the parachutist treated as a point mass at a distance from the axis of rotation. Use the formula $I = m r^2$, where $m$ is the mass of the parachutist and $r$ is the distance from the rotation axis (which is the radius of the turntable).
3. Find the total moment of inertia of the system after the parachutist lands by adding the moment of inertia of the turntable and the moment of inertia of the parachutist.
4. Use the conservation of angular momentum to find the new angular speed. The initial angular momentum can be calculated using $L_i = I_i \omega_i$, where $I_i$ is the initial moment of inertia of the turntable and $\omega_i$ is the initial angular speed. The final angular momentum is $L_f = I_f \omega_f$, where $I_f$ is the final total moment of inertia and $\omega_f$ is the final angular speed. Since angular momentum is conserved, $L_i = L_f$.
5. Solve for the final angular speed $\omega_f$ using the equation $\omega_f = \frac{L_i}{I_f}$.

Key Concepts

Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of that system remains constant. In this scenario, the initial angular momentum of the turntable and parachutist system before the parachutist lands must equal the final angular momentum after the parachutist lands, allowing us to calculate the new angular speed.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation. For a uniform disk, the moment of inertia can be calculated using the formula I = (1/2) m r², where m is the mass and r is the radius. This concept is crucial for determining the angular momentum of the turntable before and after the parachutist lands.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis, typically measured in radians per second. In this problem, we need to find the new angular velocity of the turntable after the parachutist lands, which can be derived from the conservation of angular momentum by relating the initial and final states of the system.