The motion of a particle of mass m is given by x = 0 for t < 0 s, x(t) = A sin4 t for 0 < t<(1/4)s(A>0) and x = 0 for t >(1/4) s. Which of the following statements is true?

(a) The force at t = (1/8) s on the particle is –16π² A m.

(b) The particle is acted upon by on impulse of magnitude 4π2 A m at t = 0 s and t = (1/4) s.

(c) The particle is not acted upon by any force.

(d) The particle is not acted upon by a constant force.

(e) There is no impulse acting on the particle.

2 Views | Posted 3 months ago

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Answered by

alok kumar singh | Contributor-Level 10

3 months ago

This is a Multiple Choice Questions type Questions as classified in NCERT Exemplar

Answer- a, b, d

Explanation – x=0 for t<0s

X (t)=Asin4 $π t$ for 0

X=0, for t>1/4s

For,0 $π t$

Acceleration will be, a= dv/dt=-16 $π^{2}$ Asin4 $π t$

At t= 1/8 s, a (t)=-16 $π^{2}$ Asin4 $π (\frac{1}{8})$ =-16 $π^{2}$ A

So force F =ma =-16 $π^{2}$ Am

Impulse = change in linear momentum = f (t)= -16 $π^{2}$ Am (1/4)

Impulse = change in linear momentum

=F $\times t$ =-16 $π^{2} A m \times \frac{1}{4} = - 4 π^{2} A m$

= -4 $π^{2}$ Am so clearly force depends upon a so force is also not constant .

The impulse (change in linear momentum)

At t=0 is same as t=1/4s

Clearly, force depends upon A which is not constant. Hence, force is also not constant.

This is a Multiple Choice Questions type Questions as classified in NCERT ExemplarAnswer- a, b, dExplanation – x=0 for t<0sX (t)=Asin4 <math> <mi>π</mi> <mi>t</mi> </math>  for 0<t<1 4s<="" p=""> </t<1>X=0, for t>1/4sFor,0<t<1 4s   ="" v (t)="dx/dt=" 4 πacos="" 4<span="" class="mathml" contenteditable="false"> <math> <mi>π</mi> <mi>t</mi> </math> </t<1>Acceleration will be, a= dv/dt=-16 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> Asin4 <math> <mi>π</mi> <mi>t</mi> </math> At t= 1/8 s, a (t)=-16 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> Asin4 <math> <mi>π</mi> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>8</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </math> =-16 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> ASo force F =ma =-16 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> AmImpulse = change in linear momentum = f (t)= -16 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> Am (1/4)Impulse = change in linear momentum=F <math> <mo>×</mo> <mi>t</mi> </math> =-16 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mi>A</mi> <mi>m</mi> <mo>×</mo> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mn>4</mn> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mi>A</mi> <mi>m</mi> </math> = -4 <math> <msup> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> Am so clearly force depends upon a so force is also not constant .The impulse (change in linear momentum)At t=0 is same as t=1/4sClearly, force depends upon A which is not constant. Hence, force is also not constant.

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