Question

If momentum of a particle is p = ( 2t³ - 3t² + 3t + 1 ) kgm/s. Find the force at t = 2 sec. ???? ❤❤ with proper steps ❤❤​

Answer 1

AnswEr :

The force acting on the particle is 15 N

Given :

The Momentum of the particle w.r.t time is given as :

P(t) = (2t³ - 3t² + 3t + 1 ) Kg.m/s

Explanation :

According to Newton's Second Law of Motion,

The force acting on the particle is directly proportional to the rate of change of momentum

$\large{\star \ \boxed{\boxed{\sf F = \dfrac{dp}{dt} }}}$

Now,

$\sf \: F = \dfrac{d( {2t}^{3} - {3t}^{2} + 3t + 1) }{dt} \\ \\ \longrightarrow \: \sf \: F = \dfrac{d(2 {t}^{3} )}{dt} - \dfrac{d( {3t}^{2} )}{dt} + \dfrac{d(3t) }{dt} + \dfrac{d(1)}{dt} \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf \: F(t) = 6 {t}^{2} - 6t + 3 \: N}}$

At t = 2s,the force acting on the particle would be :

$\longrightarrow \: \sf \: F = 3 \bigg(2( {2})^{2} - 2(2) + 1 \bigg) \\ \\ \longrightarrow \ \sf \: F = 3 \times 5 \\ \\ \large{ \longrightarrow \: \underline{ \boxed{ \sf \: F = 15 \: N}}}$

Answer 2

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Given :-

• Momentum of a particle is p = ( 2t³ - 3t² + 3t + 1 ) kg.m/s.

• t = 2 s

To find :-

The force.

Solution :-

We know,

$\displaystyle{\sf{F=\frac{dp}{dt} }}$

Force equals to the ratio of change in momentum to the change in time.

According to the given conditions,

$\displaystyle{\sf{F=\frac{d(2t^3-3t^2+3t+1)}{d(2)} }}\\\\\\\displaystyle{\sf{\implies F=\frac{d(2t^3)}{d(t)}+\frac{d(-3t^2)}{d(t)} +\frac{d(3t)}{d(t)}+\frac{d(1)}{d(t)} }}\\\\\\\bold{\displaystyle{\sf{\implies F(t)=6t^2-6t+3\:\:N}}}$

As it is given that t = 2 s, the force acting on the particle becomes :-

$\displaystyle{\sf{\implies F(2)=6*(2)^2-6(2)+3}}\\\\\displaystyle{\sf{\implies F=24-12+3}}\\\\\huge\boxed{\displaystyle{\sf{\implies F=15\:\: N}}}$

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