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Ch 10: Dynamics of Rotational Motion
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 10: Dynamics of Rotational MotionProblem 42a
Chapter 10, Problem 42a

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. Is the angular momentum of the block conserved? Why or why not?

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1
Identify the system: The block is revolving on a frictionless surface, attached to a cord that passes through a hole. The system is isolated in terms of horizontal forces.
Understand angular momentum conservation: Angular momentum is conserved in a system if there is no external torque acting on it. In this scenario, the only forces acting are internal (tension in the cord), and there is no external torque.
Express the initial angular momentum: The initial angular momentum (L_initial) can be expressed as L_initial = m * r_initial^2 * ω_initial, where m is the mass, r_initial is the initial radius, and ω_initial is the initial angular speed.
Express the final angular momentum: Similarly, the final angular momentum (L_final) is L_final = m * r_final^2 * ω_final, where r_final is the final radius and ω_final is the final angular speed.
Conclude on conservation: Since there is no external torque, L_initial = L_final, confirming that the angular momentum of the block is conserved as the radius changes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is given by the product of the moment of inertia and angular velocity. For a particle moving in a circle, it is calculated as L = mvr, where m is mass, v is tangential velocity, and r is the radius of the circle. In a closed system with no external torques, angular momentum is conserved.
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Conservation of Angular Momentum

The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant. This principle is crucial in analyzing systems where the radius of rotation changes, such as in this problem, where the block's radius is halved, affecting its angular speed to conserve angular momentum.
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Torque and External Forces

Torque is the rotational equivalent of force and is responsible for changes in angular momentum. It is calculated as the product of force and the lever arm distance. In this scenario, the absence of external torques (since the surface is frictionless and the cord is massless) implies that the angular momentum of the block is conserved, as no external forces are acting to change it.
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Related Practice
Textbook Question

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. Find the change in kinetic energy of the block.

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Textbook Question

A 2.00-kg rock has a horizontal velocity of magnitude 12.0 m/s when it is at point P in Fig. E10.35. If the only force acting on the rock is its weight, what is the rate of change (magnitude and direction) of its angular momentum at this instant?

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Textbook Question

A woman with mass 50 kg is standing on the rim of a large disk that is rotating at 0.80 rev/s about an axis through its center. The disk has mass 110 kg and radius 4.0 m. Calculate the magnitude of the total angular momentum of the woman–disk system. (Assume that you can treat the woman as a point.)

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Textbook Question

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. How much work was done in pulling the cord?

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Textbook Question

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. What is the new angular speed?

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Textbook Question

A hollow, thin-walled sphere of mass 12.0kg12.0\(\operatorname{kg}\) and diameter 48.0 cm48.0\(\text{ cm}\) is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by θ(t)=At2+Bt4θ(t) = At^2 + Bt^4, where A has numerical value 1.501.50 and B has numerical value 1.101.10. What are the units of the constants A and B?

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