Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

A loaded penguin sled weighing $^{80 N}$ rests on a plane inclined at angle $^{θ = 20^{0}}$ to the horizontal (Fig. 6-23). Between the sled and the plane, the coefficient of static friction is $^{0.25}$ , and the coefficient of kinetic friction is $^{0.15}$ . (a) What is the least magnitude of the force parallel to the plane, that will prevent the sled from slipping down the plane? (b) What is the minimum magnitude $^{F}$ that will start the sled moving up the plane? (c) What value of $^{F}$ is required to move the sled up the plane at constant velocity?

(a) The least magnitude of force $^{F}$ that prevents the sled moving down is $^{8.6 N}$

(b) The minimum magnitude of F that will start the sled moving up the plane is $^{46 N}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given

Weight, $^{W = 80 N}$

Coefficient of static friction, $^{μ_{s = 0.25}}$

Coefficient of kinetic friction, $^{μ_{k} = 0.15}$

Inclined angle, $^{θ = 20^{0}}$

Step 2: Determining the concept

The problem is based on Newton’s second law of motion which states that the rate of change of momentum of a body is equal in both magnitude and direction of the force acting on it. Use the Newton's 2nd law of motion along vertical and horizontal direction.

Formula:

$F_{n e t} = \sum m a$

where, F is the net force, m is mass and a is an acceleration.

Step 3: Determining the free body diagram

Free body Diagram of sled over the inclined plane:

Step 4: (a) Determining the least magnitude of the force parallel to the plane

First, from the weight, calculate the mass of the system,

$W = m g m = \frac{W}{g} = \frac{80 N}{9.81} = 8.2 kg$

By using Newton’s 2nd law along vertical direction (along y),

$\sum F_{y} = m a_{y}$

since, block is not moving upward, $a_{y} = 0$

$N - F_{g} \cos (θ) = 0 N = F_{g} \cos (θ)$

Relation between static frictional force and normal force is,

$f_{s} = μ_{s} N = N_{s} F_{g} \cos (θ)$ (i)

When sled is moving down frictional force is upward,

Now, applying Newton’s 2nd along the x direction, for fig. a,

$\sum F_{s} = m a_{x}$

since, block is not moving, localid="1654149316961" $a_{x} = 0$

$F + f_{s} - F_{g} \sin (θ) = 0 F = f_{g} \sin (θ) - f_{s}$

By using equation (i) in above equation,

$F = F_{g} \sin (θ) - μ_{s} F_{g} \cos (θ) F = F_{g} (\sin (θ) - μ_{s} \cos (θ)) = m g (\sin (20) - (0.25) \cos (20)) = 8.6 N$

Thus, the least magnitude of force $^{F}$ that prevents the sled moving down is $^{8.6 N}$

Step 5: (b) Determining the minimum magnitude of F that will start the sled moving up the plane

To find the F to start the sled moving up is, again we have to use Newton’s 2nd law along y and x direction, but in this case as the sled moving up frictional force is downward.So, refer to fig. b.

Along vertical direction (y),

$\sum F_{y} = m a_{y}$

since block is not moving upward, $a_{y} = 0$

$N - F_{g} \cos (θ) = 0 N = F_{g} \cos (θ)$

Then static frictional force,

$f_{s} = μ_{s} N = μ_{s} F_{g} \cos (θ)$

Along horizontal direction (x),

$\sum F_{x} = m a_{x}$

since, block is not moving, $a_{x} = 0$

$F - f_{s} - F_{g} \sin (θ) = 0 F = F_{g} \sin (θ) + f_{s} = F_{g} (\sin θ + μ_{s} \cos θ) = 46 N$

Hence, the minimum magnitude of F that will start the sled moving up the plane is $^{46 N}$ .

Step 6: (c) Determining thevalue of F, that is required to move the sled up the plane at constant velocity

To find $^{F}$ to move the sled up the plane at constant velocity, use kinetic frictional force,

$f_{k} = μ_{k} N = μ_{k} F_{g} \cos (θ)$

Along horizontal direction (x),

$\sum F_{x} = m a_{x}$

since, block is moving with constant velocity, $a_{x} = 0$

$F - f_{k} - F_{g} \sin (θ) = 0 F = F_{g} \sin (θ) + f_{k} = F_{g} (\sin θ + μ_{k} \cos θ) = 39 N$

Hence, the value of F, that is required to move the sled up the plane at constant velocity is $^{39 N}$

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Given

Step 2: Determining the concept

Step 3: Determining the free body diagram

Step 4: (a) Determining the least magnitude of the force parallel to the plane

Step 5: (b) Determining the minimum magnitude of F that will start the sled moving up the plane

Step 6: (c) Determining thevalue of F, that is required to move the sled up the plane at constant velocity

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: Given

Step 2: Determining the concept

Step 3: Determining the free body diagram

Step 4: (a) Determining the least magnitude of the force parallel to the plane

Step 5: (b) Determining the minimum magnitude of F that will start the sled moving up the plane

Step 6: (c) Determining thevalue of F, that is required to move the sled up the plane at constant velocity

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