Question

A body of mass 3kg is under a force which causes displacement in it given by S=2t3/3(in m). find the work done by the force in first 2 seconds​

Answer 1

Solution

• The Work Done by the force is 96 J

Given,

• Mass of the Particle (M) = 3 Kg

• Displacement,s = 2t³/3

To finD

Work Done by applied force in 2s

Firstly,

Differentiating s w.r.t to y,we get velocity function

$\sf \: v = \dfrac{ds}{dt} \\ \\ \longrightarrow \: \sf \: v = \dfrac{d( \frac{2 {t}^{3} }{3} )}{dt} \\ \\ \longrightarrow \: \sf \: v = 3 \times \dfrac{2 {t}^{2} }{3} \\ \\ \longrightarrow \: \boxed{ \boxed{\sf \: v = 2 {t}^{2} \: \: { ms}^{ - 1} }}$

Putting t = 0s would give us the initial velocity of the particle :

$\star \: \boxed{\boxed{ \sf \: u = 0 \: {ms}^{ - 1} }}$

Putting t = 2s would give us the final velocity of the particle :

$\star \: \boxed{ \boxed{ \sf v = 8 \: {ms}^{ - 1} }}$

From Work Energy Theorem,

$\tt \: W = \dfrac{1}{2} m {v}^{2} \\ \\ \longrightarrow \: \tt \: W = \dfrac{1}{2} \times 3 \times 8 \times 8 \\ \\ \huge{ \longrightarrow \: \boxed{ \boxed{ \tt{W = 96 \: J}}}}$

Answer 2

$\huge{\underline{\underline{\red{\mathfrak{AnSwEr :}}}}}$

Given :

• Mass of Body (m) = 3 kg
• Time (t) = 2 s
• S (displacement) = 2t³/3

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To Find :

• Work Done by the force applied

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Solution :

As we know that :

$\large{\boxed{\sf{v \: = \: \dfrac{ds}{dt}}}} \\ \\ \implies {\sf{v \: = \: \dfrac{d \dfrac{2t^3}{3}}{dt}}} \\ \\ \implies {\sf{v \: = \: 3 \: \times \: \dfrac{3t^2}{3}}} \\ \\ \implies {\sf{v \: = \: 2t^2}} \\ \\ {\large{\underline{\boxed{\sf{Velocity \: (v) \: = \: 2t^2 \: ms^{-1}}}}}} \\ \\ \small{\underline{\gray{\sf{\dag \: \: \: \: \: \: Put \: t \: = \: 2 \: s \: \: \: \: \: \: \dag}}}} \\ \\ \implies {\sf{v \: = \: 2 \: \times \: 2^2}} \\ \\ \implies {\sf{v \: = \: 2 \: \times \: 4}} \\ \\ \implies {\sf{v \: = \: 8}} \\ \\ {\underline{\sf{\therefore \: Velocity \: is \: 8 \: ms^{-1}}}}$

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Now, use formula for Work Done.

$\large{\boxed{\sf{W \: = \: \dfrac{1}{2}mv^2}}} \\ \\ \implies {\sf{W \: = \: \dfrac{1}{2} \: \times \: 3 \: \times \: 8^2}} \\ \\ \implies {\sf{W \: = \: \dfrac{1}{2} \: \times \: 3 \: \times \: 64}} \\ \\ \implies {\sf{W \: = \: 3 \: \times \: 32}} \\ \\ \implies {\sf{W \: = \: 96}}$

$\therefore$ Work Done by Force is 96 J