The displacement of a particle executing SHM is given by y = 5 sin 4t + (x/3). If T is the time period and the mass of the particle is 2 g, the kinetic energy of the particle when t = (T/4) is given by

Question

The displacement of a particle executing SHM is given by y = 5 sin 4t + (x/3). If T is the time period and the mass of the particle is 2 g, the kinetic energy of the particle when t = (T/4) is given by (A) 0.4 J (B) 0.5 J (C) 3 J (D) 0.3 J

Tardigrade Admin · Accepted Answer

The displacement of a particle executing SHM is given by

y = 5 sin 4 t + \frac{x}{3} .

......(i)
Velocity of particle

\frac{d y}{d t} = \frac{5 d}{d t} sin 4 t + \frac{π}{3}

= 5 cos 4 t + \frac{π}{3}

= 20 cos 4 t + \frac{π}{3}

Velocity at

t = \frac{T}{4}

\frac{d y}{d t}_{t = \frac{T}{4}} = 20 cos 4 \times \frac{T}{4} + \frac{π}{3}

Or

u = 20 cos T + \frac{π}{3}

....(ii)
Now, putting value of T in Eq. (ii), we get

u = 20 cos \frac{π}{2} + \frac{π}{3}

= - 20 sin \frac{π}{3}

= - 20 \times \frac{3}{2}

= - 10 \times 3

The kinetic energy of particle,

K E = \frac{1}{2} m u^{2}

∵ m = 2 g = 2 \times 1 0^{- 3} k g

= \frac{1}{2} \times 2 \times 1 0^{- 3} \times - 10 \overline{3}^{2}

= 1 0^{- 3} \times 100 \times 3

= 3 \times 1 0^{- 1}

K E = 0.3 J

Q. The displacement of a particle executing SHM is given by y=5sin4t+3x​. If T is the time period and the mass of the particle is 2g, the kinetic energy of the particle when t=4T​ is given by

0.4 J

0.5 J

3 J

0.3 J

Q. The displacement of a particle executing SHM is given by
$y = 5 sin 4 t + \frac{x}{3} .$
If $T$ is the time period and the mass of the particle is $2 g$ , the kinetic energy of the particle when $t = \frac{T}{4}$ is given by