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Chapter 7: Q.54 (page 180)

In $F I G U R E C P 7.54$ , find an expression for the acceleration of m1. The pulleys are massless and frictionless.
Hint: Think carefully about the acceleration constraint.

Short Answer

Expert verified

The acceleration of block $m_{1}$ is given by $a_{1} = \frac{2 m_{2} g}{(2 m_{1} + m_{2})}$

Step by step solution

01

Given information

Mass of the block $1$ is $m_{1}$

mass of the block $2$ is $m_{2}$

The pulleys are massless and frictionless

The acceleration of the block $1$ is localid="1649864857767" $a_{1}$

The acceleration of the block $2$ is $a_{2}$

02

Explanation

Consider the Forces of two masses given in the $f i g u r e C P 7.54$

For block of mass $m_{1}$

For block of mass $m_{2}$

$m_{2} g - T_{2} = m_{2} a_{2}$

Considering the pulley is massless, $鈭� T_{1} = T_{2}$

Let $a_{2} = a$ $鈭� T = 2 m_{1} a$

$鈭� a_{1} = 2 a$ acceleration constraint

On substituting the above equations on block equation, we get

$m_{2} g - 2 m_{1} a = m_{2} a (2 m_{1} + m_{2}) a = m_{2} g a = \frac{m_{2} g}{(2 m_{1} + m_{2})}$

Therefore, the acceleration for $m_{1}$ is given by,

$a_{1} = \frac{2 m_{2} g}{(2 m_{1} + m_{2})}$

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Most popular questions from this chapter

The cable cars in San Francisco are pulled along their tracks by an underground steel cable that moves along at $9.5 m p h .$ The cable is driven by large motors at a central power station and extends, via an intricate pulley arrangement, for several miles beneath the city streets. The length of a cable stretches by up to $100 f t$ during its lifetime. To keep the tension constant, the cable passes around a $1.5 m$ diameter 鈥渢ensioning pulley鈥� that rolls back and forth on rails, as shown inFIGURE $100 f t$ . A $2000 k g$ block is attached to the tensioning pulley鈥檚 cart, via a rope and pulley, and is suspended in a deep hole. What is the tension in the cable car鈥檚 cable?

A $1000 k g$ car pushes a $2000 k g$ truck that has a dead battery. When the driver steps on the accelerator, the drive wheels of the car push against the ground with a force of $4500 N$ . Rolling friction can be neglected.

a. What is the magnitude of the force of the car on the truck?

b. What is the magnitude of the force of the truck on the car?

The sled dog in the figure drags sleds A and B across the snow. The coefficient of friction between the sleds and the snow is $0.10 .$ If the tension in rope $1$ is $150 N,$ what is the tension in rope $2 ?$

$F I G U R E E X 7.16$ shows two $1.0 k g$ blocks connected by a rope. A second rope hangs beneath the lower block. Both ropes have a mass of $250 g$ . The entire assembly is accelerated upward at $3.0 m / s^{2}$ by force $\overset{鈫�}{F}$

a. What is F?

b. What is the tension at the top end of rope $1$ ?

c. What is the tension at the bottom end of rope $1$ ?

d. What is the tension at the top end of rope $2$ ?

In case a, in the figure block A is accelerated across a frictionless table by a hanging $10 N$ weight $(1.02 k g)$ . In case b, block A is accelerated across a frictionless table by a steady $10 N$ tension in the string. The string is massless, and the pulley is massless and frictionless. Is A鈥檚 acceleration in case b greater than, less than, or equal to its acceleration in case a? Explain.

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