Question

The magnitude of force (in $ N $ ) acting on a body varies with time t (in $ \mu s $ ) is shown in the Figure. AB, BC and CD are straight line segments. The magnitude of total impulse of the force acting on the object from  $ t=4\mu s $ to $ t=16\mu s $  is given as______\[Ns\].

$A{{.10}^{-3}}s$

$B{{.10}^{-2}}s$

$C{{.10}^{-4}}s$

$D.5\times {{10}^{-4}}s$

$E.5\times {{10}^{-3}}s$

Answer 1

Hint:  Impulse is a concept based on momentum. An impulse is defined as the resultant force on a body times the time period over which this force is experienced. Impulse can be calculated from a force-time graph by taking the area under the graph in consideration.

Complete answer:

In the question, a graph is given which is having force on its y axis and time on its x axis. We have to find the impulse of the body from $ t=4\mu s $ to $ t=16\mu s $ time interval.

As we all know,

The impulse of a body can be expressed as,

$I=F\times t $

Where $ F $ is the force acting and $ t $ is the time taken.

When we look into the graph, we can see that the area under can be given as the equation,

 area=F\times t $

Therefore the area under a force time graph will be equal to the value of impulse.

From the graph,

Impulse= area of EBCD.

$areaEBCD=areaEBCF+\Delta FCD $

Therefore the impulse can be written as,

$I=\left( \dfrac{BE+FC}{2}\times EF \right)+\dfrac{1}{2}FC\times FD $

Substituting the values in it,

$I=\left( \dfrac{200+1800}{2}\times 2\times {{10}^{-6}} \right)+\left[ \dfrac{1}{2}\times 800\times 10\times {{10}^{-6}} \right] $

We have to simplify this, which will give,

$I=\left( 1000+4000 \right)\times {{10}^{-6}}Ns$

$I=5000\times {{10}^{-6}}Ns$

$I=5\times {{10}^{-3}}Ns$

Therefore the impulse acting over the body has been found out.

So, the correct answer is “Option E”.

Note: We can develop a direct connection between how a force is acting on a body over a period of time and the motion of the body as well. This is because of the impulse momentum theorem. When a collision happens, a body experiences a force for a certain period of time that will result in its mass undergoing a variation in velocity due to change in momentum. This is referred to as the impulse-momentum change theorem.