Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 10: Dynamics of Rotational Motion

Problem 35a

Chapter 10, Problem 35a

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. At this instant, what are the magnitude and direction of its angular momentum relative to point O?

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Identify the given values: mass of the rock (m) is 2.00 kg, velocity (v) is 12.0 m/s, distance from point O to point P (r) is 8.00 m, and the angle between the line OP and the horizontal is 36.9 degrees.

Recall the formula for angular momentum (L) relative to a point: L = r * m * v * sin(θ), where θ is the angle between the position vector and the velocity vector.

Determine the angle θ between the position vector and the velocity vector. Since the velocity is horizontal and the position vector makes an angle of 36.9 degrees with the horizontal, θ = 90 degrees - 36.9 degrees.

Substitute the known values into the angular momentum formula: L = 8.00 m * 2.00 kg * 12.0 m/s * sin(53.1 degrees).

Calculate the sine of 53.1 degrees and multiply all the values to find the magnitude of the angular momentum. The direction of the angular momentum is perpendicular to the plane formed by the position vector and the velocity vector, following the right-hand rule.

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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 cross product of the position vector and the linear momentum vector. For a point mass, it is calculated as L = r × p, where r is the position vector from the point of rotation to the object, and p is the linear momentum (mass times velocity).

Cross Product

The cross product is a vector operation used to find a vector perpendicular to two given vectors in three-dimensional space. It is essential in calculating angular momentum, as it determines the direction of the angular momentum vector. The magnitude of the cross product is given by |A × B| = |A||B|sin(θ), where θ is the angle between vectors A and B.

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 rotational motion is involved, as it allows for the prediction of future states of the system based on initial conditions.

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Conservation of Angular Momentum

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Textbook Question

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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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