question_answer 50)

                  The motion of a particle of mass m is given by \[x=0\] for \[t<0\text{ }s,\text{ }x(t)=A\text{ }\sin \text{ }4\pi \,t\] for \[0<t<(1/4)s\text{ }(A>o),\] and  \[x=0\] for \[t>(1/4)s\]. Which of the following statements is true?                 (a) The force at \[t=(1/8)s\] s on the particle is \[-16{{\pi }^{2}}\,A\,m\] .                 (b) The particle is acted upon by an impulse of magnitude \[4\,{{\pi }^{2}}\,A\,m\] 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.

Answer:

                  (a, b, d) \[x(t)=A\sin 4\,\pi t\]                 \ \[\upsilon (t)=4\pi \,A\cos \text{ }4\pi t\] and                 \[a(t)=16{{\pi }^{2}}A\text{ }\sin \text{ }4\pi t\]                 When \[t=\frac{1}{8}\,s,\,\,a=-16\,{{\pi }^{2}}A\]                 \[\therefore \]\[F=\,ma\,=-16\,{{\pi }^{2}}\,Am\]                 Impulse \[=Ft=16{{\pi }^{2}}\,Am\times \frac{1}{4}=4{{\pi }^{2}}\,Am\]                 Since F depends on A (not constant), so F changes.