CP A small block on a frictionless, horizontal surface has a mass of 0.0250 kg. It is attached to a massless cord passing through a hole in the surface (Fig. E10.40). The block is originally revolving at a distance of 0.300 m from the hole with an angular speed of 2.85 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to 0.150 m. Model the block as a particle. (a) Is the angular momentum of the block conserved? Why or why not?

Verified step by step guidance

1. Identify the system and the forces acting on it. In this scenario, the system consists of the block and the cord. The only external force acting on the system is the tension in the cord, which is internal to the system. There are no external torques acting on the system since the tension force acts along the line of the radius and thus has no lever arm to create a torque about the center.
2. Understand the concept of angular momentum conservation. Angular momentum is conserved in a system if there is no net external torque acting on it. Since the only forces acting are internal and along the radius of the motion, there is no external torque.
3. Apply the law of conservation of angular momentum. The initial angular momentum can be calculated using the formula $L_i = I_i \omega_i$, where $I_i$ is the initial moment of inertia and $\omega_i$ is the initial angular speed.
4. Calculate the final angular momentum using the formula $L_f = I_f \omega_f$, where $I_f$ is the final moment of inertia and $\omega_f$ is the final angular speed. Since angular momentum is conserved, $L_i = L_f$.
5. Relate the initial and final conditions to find the final angular speed. Using the conservation of angular momentum and the relationship between the initial and final moments of inertia (which depend on the radius of rotation), solve for the final angular speed $\omega_f$.

Key Concepts

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is defined as the product of the object's moment of inertia and its angular velocity. For a particle moving in a circular path, it can be expressed as L = mvr, where L is angular momentum, m is mass, v is linear velocity, and r is the radius of the circular path. Understanding angular momentum is crucial for analyzing rotational systems and determining whether it is conserved in a given scenario.

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 the context of the problem, if the only forces acting on the block are internal (like tension in the cord), then the angular momentum before and after the radius change should be compared to determine if it is conserved. This principle is fundamental in understanding how systems behave when their configuration changes.

Frictionless Surface

A frictionless surface is an idealized concept where no frictional forces oppose the motion of an object. In this scenario, the block can revolve without any energy loss due to friction, allowing for a clear analysis of its angular momentum. This simplification is important for applying the conservation laws effectively, as it ensures that the only forces to consider are those related to the tension in the cord and the gravitational forces acting on the system.