Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

Answers without the blur.

Just sign up for free and you're in.

Short Answer

In a playground, there is a small merry-go-round of radius 1.20 m and mass 180 kg. Its radius of gyration (see Problem 79 of Chapter 10) is 91.0 cm. A child of mass 44.0 kg runs at a speed of 3.00 m/s along a path that is tangent to the rim of the initially stationary merry-go-round and then jumps on. Neglect friction between the bearings and the shaft of the merry-go-round. Calculate (a) the rotational inertia of the merry-go-round about its axis of rotation, (b) the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round, and (c) the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round.

Rotational Inertia of merry go round about its axis of rotation is 149 kg.m².
The magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round is 158 kg.m²/s.
The angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round is 0.744 rad/s².

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given

$m = 180 kg k = 91 cm M = 44 kg r = 1.20 m v = 3 m / s$

Step 2: Determining the concept

Use the formula for rotational inertia in terms of mass and radius of gyration to find the rotational inertia. Using the formula for angular momentum in terms of mass, velocity and radius, find the angular momentum. Finally, use conservation of angular momentum to find angular velocity. According to the conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula are as follow:

$I = m \times k^{2} L = m \times v \times r$

where,r is radius, v is velocity, m is mass, L is angular momentum, l is moment of inertia and r is radius of gyration.

Step 3: Determining the rotational Inertia of merry go round about its axis of rotation

(a)

To find rotational inertia, use the following formula:

$I = m \times k^{2} I = 180 \times 0 . 91^{2} I = 149 kg . m^{2}$

Hence, rotational Inertia of merry go round about its axis of rotation is 149 kg.m².

Step 4: Determining the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round

(b)

Now, to find angular momentum,

$L = M \times r \times v L = 44 \times 3 \times 1.2$

So,

data-custom-editor="chemistry" $L = 158 kg . m^{2} / s$

Hence, the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round is 158 kg.m²/s.

Step 5: Determining the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round

(c)

Now, to find angular velocity, use conservation of momentum,

L_{f} = L_{child} L_{f} = (I + {mr}^{2}) ω L_{child} = mvr mvr = (I + {mr}^{2}) ω ω = \frac{mvr}{I + {mr}^{2}}

So,

$ω = \frac{158}{149 + 44 + 1 . 2^{2}} = 0.744 rad / s ω = 0.744 rad / s$

Hence, the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round is 0.744 rad/s².

Therefore, use the formula for rotational inertia in terms of mass and radius of gyration to find the rotational inertia. Using the formula for angular momentum in terms of mass, velocity and radius, the angular momentum can be found. Finally, conservation of angular momentum can be used to find angular velocity.

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Given

Step 2: Determining the concept

Step 3: Determining the rotational Inertia of merry go round about its axis of rotation

Step 4: Determining the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round

Step 5: Determining the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Given

Step 2: Determining the concept

Step 3: Determining the rotational Inertia of merry go round about its axis of rotation

Step 4: Determining the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round

Step 5: Determining the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics