Ch 12: Rotation of a Rigid Body

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

Problem 26

Chapter 12, Problem 26

A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid, massless rod. The rod is rotating cw about its center of mass at 20 rpm. What net torque will bring the balls to a halt in 5.0 s?

Verified step by step guidance

Step 1: Calculate the center of mass of the system. Since the rod is massless, the center of mass depends only on the masses of the balls. Use the formula for the center of mass:

x = \frac{m_{1} x_{1} + m_{2} x_{2}}{m_{1} + m_{2}}

, where

x_{1}

and

x_{2}

are the positions of the masses relative to one end of the rod.

Step 2: Determine the moment of inertia of the system about the center of mass. Use the formula for the moment of inertia:

I = m_{1} r^{1} + m_{2} r^{2}

, where

r_{1}

and

r_{2}

are the distances of the masses from the center of mass.

Step 3: Convert the angular velocity from rpm to rad/s. Use the conversion factor:

(1 rpm) = \frac{2 π}{60} rad/s

. Multiply the given angular velocity (20 rpm) by this factor to find the angular velocity in rad/s.

Step 4: Calculate the angular deceleration required to bring the system to a halt in 5.0 s. Use the kinematic equation:

α = \frac{ω - 0}{t}

, where

ω

is the initial angular velocity, and

t

is the time.

Step 5: Calculate the net torque required to achieve this angular deceleration. Use the rotational analog of Newton's second law:

τ = I α

, where

I

is the moment of inertia and

α

is the angular deceleration.

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Key Concepts

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

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). It determines how effectively a force can cause an object to rotate about an axis. In this scenario, understanding torque is essential to calculate the net torque required to stop the rotating system.

Moment of Inertia

Moment of inertia quantifies an object's resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. For the two balls connected by a rod, the moment of inertia can be calculated by considering the masses of the balls and their distances from the center of mass. This concept is crucial for determining how much torque is needed to bring the system to a halt.

Angular Deceleration

Angular deceleration refers to the rate at which an object's angular velocity decreases over time. It is the rotational equivalent of linear acceleration and is necessary to calculate the required torque to stop the rotation. In this problem, knowing the initial angular velocity and the time to stop allows us to find the angular deceleration needed to halt the system.

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