Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 4, Problem 122P

(a)

To determine

The acceleration of each blocks and tension in the cord.

(a)

Expert Solution
Check Mark

Answer to Problem 122P

The magnitude of acceleration of each blocks and tension in the cord is respectively (m2μkm1m1+m2)g and (μk+1)m1m2m1+m2g.

Explanation of Solution

Case 1 (mass m1).

Write the expression for the equilibrium condition in y-direction (Vertical).

N=m1g (I)

Here, N is the normal force on the mass m1 and g is the acceleration due to gravity.

Write the expression for the equilibrium condition in x-direction (Horizontal).

Tμkm1g=m1ax . (II)

Here, ax is the acceleration of mass m1 in the horizontal direction and T is the tension in the cord.

Rewrite the above equation in terms of ax.

ax=Tm1μkg . (III)

Case 2 (mass m2).

Write the expression for the equilibrium condition in y-direction (Vertical).

Tm2g=m2ay (IV)

Here, ay is the acceleration of mass m2.

Write the above expression in terms of ay.

ay=Tm2g . (V)

Since both acceleration is equal in magnitude and opposite in direction.

Write the expression for the relation between acceleration of two blocks.

ax=ay (VI)

Substitute (III) and (V) in the above equation to calculate T.

Tm1μkg=Tm2+gT(1m1+1m2)=g(μk+1)T(m1+m2m1m2)=g(μk+1)T=(μk+1)m1m2m1+m2g (VII)

Substitute (VI) in (v) to calculate the acceleration ay.

ay=1m2((μk+1)m1m2m1+m2g)g=(μk+1)m1m1+m2gg=g(μkm1+m1m1+m21)=μkm1m2m1+m2g

Substitute the above expression in (VI) to calculate ax.

ax=(μkm1m2m1+m2g)ax=(m2μkm1m1+m2)g . (VIII)

Conclusion:

Therefore, the magnitude of acceleration of each blocks and tension in the cord is respectively (m2μkm1m1+m2)g and (μk+1)m1m2m1+m2g.

(b)

To determine

The magnitude of acceleration and tension for the cases m1<<m2, m1>>m2 and m1=m2.

(b)

Expert Solution
Check Mark

Answer to Problem 122P

The magnitude of acceleration and tension for the cases m1<<m2, m1>>m2 and m1=m2 is respectively is zero or free fall and g, m2g and 0, (μk+1)m2g2 and ((1μk)2)g.

Explanation of Solution

Consider the equation (VII) and (VIII).

Case 1 m1<<m2.

T=(μk+1)m1m2m1+m2g=(μk+1)m1g(1+m1m2)(μk+1)m1g

Since μk is too small, μk could possible to neglect.

Thus, the tension in the cord is m1g and is negligible might be considered as free-fall.

ax=(m2μkm1m1+m2)g=(m2m1μk(1+m2m1))g

Since 1+m2m1 is m2m1 and μk is too small, μk could possible to neglect.

Thus, the acceleration of the block is g.

Case 2 m1>>m2.

T=(μk+1)m1m2m1+m2g=(μk+1)m2g(1+m2m1)(μk+1)m2g

Since μk is too small, μk could possible to neglect.

Thus, the tension in the cord is m2g.

ax=(m2μkm1m1+m2)g=(m2m1μk(1+m2m1))g

Since 1+m2m1 is m2m1 and μk is too small, μk could possible to neglect.

Thus, the acceleration of the block is 0.

Case 3 m1=m2.

Substitute m1=m2 in (VII) and (VIII) to calculate T and ax.

T=(μk+1)m2m2m2+m2g=(μk+1)m22g(2m2)=(μk+1)m2g2

ax=(m2μkm2m2+m2)g=(m2(1μk)2m2)g=((1μk)2)g

Conclusion:

Therefore, the magnitude of acceleration and tension for the cases m1<<m2, m1>>m2 and m1=m2 is respectively is zero or free fall and g, m2g and 0, (μk+1)m2g2 and ((1μk)2)g.

(c)

To determine

The magnitude of m2 and tension in the cord.

(c)

Expert Solution
Check Mark

Answer to Problem 122P

The magnitude of m2 and tension in the cord is respectively μkm1 and m2g.

Explanation of Solution

Consider the equation (VII) and (VIII).

In the case of constant velocity acceleration should be zero.

Equate (VIII) to zero to calculate m2.

(m2μkm1m1+m2)g=0m2μkm1=0m2=μkm1

Substitute the above equation for m2 in (VII) to calculate T.

T=(μk+1)m1m2m1+(μkm1)g=(μk+1)m1m2m1(1+μk)g=m2g

Conclusion:

Therefore, the magnitude of m2 and tension in the cord is respectively μkm1 and m2g.

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