Subjective Type

In a simple Atwood machine, two unequal masses $${m_1}$$ and $${m_2}$$ are connected by a string going over a clamped light smooth pulley. In a typical arrangement shown in figure, $${m_1}=300 g$$ and $${m_2}=600 g$$. The system is released from the rest.Find the distance traveled by the first block in the first two seconds.

Solution

$$m_{1}=0.3\ kg, m_{2}=0.6\ kg$$
$$T-(m_{1}g+m_{2}a)=0$$ ... (i)
$$\Rightarrow T=m_{1}g+m_{1}a$$
$$T+m_{2}a-m_{2}g=0$$ .... (ii)
$$\Rightarrow T=m_{2}g-m_{2}a$$
From equation (i) and equation (ii)
$$m_{2}g+m_{1}a+m_{2}a-m_{2}g=0$$,
from $$(i)$$
$$\Rightarrow a(m_{1}+m_{2})=g(m_{2}-m_{1})$$
$$\Rightarrow a=f\left(\dfrac{m_{2}-m_{1}}{m_{1}+m_{2}}\right)=9.8\left(\dfrac{0.6-0.3}{0.6+0.3}\right)=3.266\ ms^{-2}$$
a) $$t=2\ sec$$ acceleration $$=3.266\ ms^{-2}$$
Initial velocity $$u=0$$
So, distance travelled by the body is $$S=ut+1/2 at^{2}\Rightarrow 0+1/2(3.266)2^{2}=6.5\ m$$

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