Question

4. The position (x) of a body moving along x-axis
at time (t) is given by x = 3t', where x is in
metre and t is in second. If mass of body is 2
kg, then find the instantaneous power delivered
to body by force acting on it at t = 4s :-
(1) 288 W
(2) 280 W
(3) 290 W
(4) None

Answer 1

Answer:

instantaneous power=W/t

W=fd

Nd

X=3t

F=ma

instantaneous power=1152/4

=288watt

Answer 2

$\huge{\bold{\underline{\underline{\tt Correct \: question:-}}}}$

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The position (x) of a body moving along x-axis at time (t) is given by x = 3t², where x is in metre and t is in second. If mass of body is 2 kg, then find the instantaneous power delivered to body by force acting on it at t = 4s.

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$\huge{\bold{\underline{\underline{....Answer....}}}}$

$\huge{\bold{\underline{Given:-}}}$

• X = 3t² m.
• Mass of the body (m) = 2 kg.
• Time period = 4 seconds.

$\huge{\bold{\underline{Explanation:-}}}$

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As we know,

Power is rate of doing work.

Now, It is expressed as :-

$\large{\boxed{\tt P = \dfrac{W}{t}}}$

But for instantaneous power,

The time intervals should be small, Then,

$\large{\boxed{\tt P_{inst} = \dfrac{dw}{dt}}}$

Now,

Work done = Force × Displacement.

I.e. W = F.S.

Substituting it,

$\large{\tt P_{inst} = \dfrac{d(F.s)}{dt}}$

As Force is constant it cannot be differentiated,

Then,

$\large{\tt P_{inst} = F.\dfrac{ds}{dt}}$

Now, as we know, ${\tt \left[\dfrac{ds}{dt} = v \right]}$

Substituting it,

$\large{\tt P_{inst} = F \times v}$

$\large{\boxed{\tt P_{inst} = F.v}}$

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Finding the Force,

As we know,

The relation between position and time I.e.

$\large{\boxed{\tt x = 3t^2}}$

Now, differentiating it w.r.t time. we will get velocity,

$\large{\tt v = \dfrac{d(x)}{dt}}$

$\large{\tt v = \dfrac{d(3t^2)}{dt}}$

$\large{\tt v = 6t \: m/s \: ----(1)}$

$\large{\boxed{\tt v = 6t \: m/s}}$

Now, Again differentiating velocity w.r.t time, we will get acceleration.

$\large{\tt a = \dfrac{d(v)}{dt}}$

$\large{\tt a = \dfrac{d(6t)}{dt}}$

$\large{\tt a = 6 \: m/s^2 \: ----(2)}$

$\large{\boxed{a = 6 \: m/s}}$

From Newton's second law of motion,

$\large{\boxed{\tt F = ma}}$

Substituting the values,

$\large{\tt F = 2 \times 6}$

$\large{\boxed{\tt F = 12 \: N}}$

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Substituting it in the Instantaneous Power formula,

$\large{\boxed{\tt P_{inst} = F.v}}$

Substituting the values,

[From equation (1) substituting the velocity]

$\large{\tt P_{inst} = 12 \times 6t}$

Now, As the question specifies The time to be taken as 4 seconds.

Now,

$\large{\tt P_{inst} = 12 \times [6 \times 4]}$

$\large{\tt P_{inst} = 12 \times 24}$

$\huge{\boxed{\boxed{\tt P_{inst} = 288 \: W}}}$

So, the Instantaneous Power of the body at 4 seconds is 288 Watts.

So, the correct option is (1).

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