Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 11, Problem 64P

(a)

To determine

The gravitational force on the particle of mass m due to the five objects

(a)

Expert Solution
Check Mark

Answer to Problem 64P

The gravitational force on the particle of mass m due to the five particles is 9.7×108Nj^.

Explanation of Solution

Given data:

Following figure shows the distribution of masses on the semi-circular arc of radius R

Physics for Scientists and Engineers, Chapter 11, Problem 64P , additional homework tip  1

Formula Used:

The expression for the gravitational force between the masses m and M is

  F1=GMmR2i^

Here, G is the gravitational constant, M is the mass of the one object, m is the mass of the other object and R is the radius between the two objects. Here, i^ represents the gravitational force along x -direction.

Calculation:

Radius of the arc is 10cm . Convert the distance from cm into m .

  R=(10.0cm)(1m102cm)=10×10-2m

Choose the unit vector i^ and j^ along the positive x -axis and positive y -axis respectively.

The gravitational force between the masses m and M which makes an angle 45o with the positive x -axis is:

  F2=(GMmR2)cos45oi^+(GMmR2)sin45oj^

The gravitational force between the masses m and M which is on positive y-axis is:

  F3=GMmR2j^

The gravitational force between the masses m and M which makes an angle 135o with the positive x -axis is:

  F4=(GMmR2)cos135oi^+(GMmR2)sin135oj^

The gravitational force between the masses m and M which is on lest side x -axis is,

  F5=GMmR2i^

Hence, net force on the particle of mass m is,

  F=F1+F2+F3+F4+F5=GMmR2 i ^ +(GMmR2)cos45o i ^ +(GMmR2)sin45o j ^ +GMmR2 j ^ -(GMmR2)cos45o i ^ +(GMmR2)sin45o j ^ -GMmR2 i ^=(2(GMmR2)sin45o+GMmR2) j ^ N=GMmR2(2sin45o+1) j ^ N

Substitute the values and solve:

  F=(6.67×10-11N.m2/kg2)(3.0kg)(2.0kg)(0.1m)2(2sin45o+1) j ^ N=9.7×10-8 j ^ N

Conclusion:

Therefore, the gravitational force on the particle of mass m due to the five particles is 9.7×108N j ^ .

(b)

To determine

The gravitational field at center of curvature of the arc.

(b)

Expert Solution
Check Mark

Answer to Problem 64P

The gravitational field at center of curvature of the arc is 4.85×108N/kg j ^ .

Explanation of Solution

Given data:

Following figure shows the distribution of masses on the semi-circular arc of radius R

Physics for Scientists and Engineers, Chapter 11, Problem 64P , additional homework tip  2

Formula Used:

Gravitational field at a point is:

  g=Fgm

Here, net gravitational force is Fg and mass is m

Calculation:

The gravitational field at the center of curvature of the arc can be determined as follows.

  g=Fgm

Substitute 9.7×108Nj^ for Fg and 2.0kg for m

  g=9.7×108Nj^2.0kg=4.85×108N/kgj

Conclusion:

The gravitational field at center of curvature of the arc is 4.85×108N/kg j ^ .

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Chapter 11 Solutions

Physics for Scientists and Engineers

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