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Physics Principles with Applications
Found in: Page 134
Physics Principles with Applications

Physics Principles with Applications

Book edition 7th
Author(s) Douglas C. Giancoli
Pages 978 pages
ISBN 978-0321625922

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Short Answer

Four 7.5-kg spheres are located at the corners of a square of side 0.80 m. Calculate the magnitude and direction of the gravitational force exerted on one sphere by the other three.

The magnitude and direction of the gravitational force is F0=1.1×10-8 N and ϕ=45° .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1. Given Data

The mass of spheres is m=7.5 kg.

The side of the square is d=0.8 m.

Step 2. Understanding the net force on the sphere

In this problem, the net forces on the sphere in the horizontal as well as vertical direction will be equal due to symmetry.

Step 3. Estimating the magnitude of net force exerted on the sphere

The free body diagram of the sphere is as follows:

The relation of net force in the horizontal direction is given by,

Fx=FR+FcosθFx=Gm2d2+Gm22d2cosθ

Here, G is the gravitational constant, FR is the force on the right direction, F is the force on the diagonal and θ is the angle made by each sphere whose value is θ=45°.

On plugging the values in the above relation.

Fx=Gm2d2+Gm22d2cos45°Fx=Gm2d2+Gm22d212Fx=Gm2d21+122

The force on the vertical direction will be having the same magnitude because of the symmetry.

Step 4. Estimating the resultant of the net force exerted on the sphere

The relation of resultant of the net force given by,

F0=Fx2+Fy2F0=Fx2+Fx2F0=2Fx2F0=2Fx

On plugging the values in the above relation.

F0=2Gm2d21+122F0=26.67×10-11 N·m2/kg27.5 kg20.8 m21+122F0=1.1×10-8 N

Thus, F0=1.1×10-8 N is the required magnitude of the net force.

Step 5. Estimating the direction of net force exerted on the sphere

The relation of direction of the net force given by,

tanϕ=FyFx

On plugging the values in the above relation.

tanϕ=Gm2d21+122Gm2d21+122ϕ=45°

Thus, ϕ=45° is the required direction of the net force.

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