Book edition 7th

Author(s) Douglas C. Giancoli

Pages 978 pages

ISBN 978-0321625922

Answers without the blur.

Just sign up for free and you're in.

Short Answer

A block with mass \(M = 6.0\;{\rm{kg}}\) rests on a frictionless table and is attached by a horizontal spring \(\left( {k = 130\;{\rm{N/m}}} \right)\) to a wall. A second block, of mass \(m = 1.25\;{\rm{kg}}\), rests on top of \(M\). The coefficient of static friction between the two blocks is \(0.30\). What is the maximum possible amplitude of oscillation such that \(m\) will not slip off \(M\)?

The maximum possible amplitude of oscillation is 0.16 m.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Concept of acceleration applied on frictional surface

Acceleration applied on the frictional surface is calculated using the coefficient of static friction and normal reaction, which ultimately equivalent to the weight of the object.

Given data

The mass of the block is \(M = 6.0\;{\rm{kg}}\).

The mass of the second block is \(m = 1.25\;{\rm{kg}}\).

The spring stiffness constant is \(k = 130\;{\rm{N/m}}\).

The coefficient of static friction is \({\mu _{\rm{s}}} = 0.30\).

Calculation of maximum amplitude

In the stationary situation, static friction will apply in between both of the blocks. This frictional force causes block to move upon the bigger block M. Thus, equating both of the forces due to acceleration and static friction as:

\(\begin{aligned}{c}F = {F_{\rm{f}}}\\m{a_{{\rm{max}}}} = {\mu _{\rm{s}}}mg\\{a_{{\rm{max}}}} = {\mu _{\rm{s}}}g\end{aligned}\)

Here, \({F_{\rm{f}}}\) is the frictional force, F is the force due to acceleration, \({a_{{\rm{max}}}}\) is the maximum acceleration and \(g\) is the acceleration due to gravity.

Now, for this second block to stay stationary on the bigger block without slipping, maximum acceleration of bigger block of mass \(M\) is also \({a_{{\rm{max}}}} = {\mu _{\rm{s}}}g\). However, due to connection with spring, system of block will perform simple harmonic motion, then the maximum acceleration is calculated as:

\(\begin{aligned}{c}{a_{{\rm{max}}}} = {\omega ^2}A\\ = \frac{k}{{m + M}}A\end{aligned}\)

Here, \(\omega \) is the angular frequency and A is the amplitude.

Equating this equation with the previous expression of maximum acceleration as:

\(\begin{aligned}{c}{\mu _{\rm{s}}}g &= \frac{k}{M}A\\A &= \frac{{{\mu _{\rm{s}}}g}}{k}\left( {m + M} \right)\\A &= \frac{{\left( {0.30} \right)\left( {9.8\;{\rm{m/}}{{\rm{s}}^2}} \right)}}{{\left( {130\;{\rm{N/m}}} \right)}}\left( {6.0\;{\rm{kg}} + 1.25\;{\rm{kg}}} \right)\\A &= 0.16\;{\rm{m}}\end{aligned}\)

Hence, the maximum amplitude of the oscillations is 0.16 m.

Find Study Materials for

Create Study Materials

Select your language

Physics Principles with Applications

Answers without the blur.

Short Answer

Step by Step Solution

Concept of acceleration applied on frictional surface

Given data

Calculation of maximum amplitude

Most popular questions for Physics Textbooks

A satellite in circular orbit around the Earth moves at constant speed. This orbit is maintained by the force of gravity between the Earth and the satellite, yet no work is done on the satellite. How is this possible?

(a) No work is done if there is no contact between objects.

(b) No work is done because there is no gravity in space.

(c) No work is done if the direction of motion is perpendicular to the force.

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Physics Principles with Applications

Answers without the blur.

Short Answer

Step by Step Solution

Concept of acceleration applied on frictional surface

Given data

Calculation of maximum amplitude

Most popular questions for Physics Textbooks

A satellite in circular orbit around the Earth moves at constant speed. This orbit is maintained by the force of gravity between the Earth and the satellite, yet no work is done on the satellite. How is this possible?

(a) No work is done if there is no contact between objects.

(b) No work is done because there is no gravity in space.

(c) No work is done if the direction of motion is perpendicular to the force.

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Famous Physicists

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