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Question

Prove Work Energy Theorem by a variable force.


Solution

$$\Delta W=F(x)\Delta x$$
$$W=\sum _{ x_ i }^{ x_f }{F(x)\Delta x  } $$
$$W=\lim_{\Delta x\rightarrow 0}{\sum_{x_i}^{x_f}{F(x)\Delta x}}$$
or,
$$W=\int_{x_i}^{x_f}{F(x)dx}$$
$$K=\cfrac{1}{2}mv^2$$
$$\cfrac{dK}{dt}=m\cfrac{dv}{dt}\cdot v$$
$$\Rightarrow \cfrac{dK}{dt}=F\cfrac{dx}{dt}$$
$$\Rightarrow \Delta K=\int_{x_i}^{x_f}{F\cdot dx}$$

978122_1076914_ans_9fd86d830cf3439f844be82fc53fe27d.png

Physics

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