The kinetic energy of a particle moving along a circle of radius $R$ depends on the distance covered $s$ as $T = a s^{2}$ , where $a$ is a constant. Find the force acting on the particle as a function of $s$ .

Open in App

Solution

We have $T = \frac{1}{2} m v^{2} = a s^{2}$ or $v^{2} = \frac{2 a s^{2}}{m}$ (1)
Differentiating equation (1) w.r.t. time
$2 v w_{t} = \frac{4 a s}{m} v$ or, $w_{t} = \frac{2 a s}{m}$ (2)
Hence net acceleration of the particle
$w = \sqrt{w_{t}^{2} + w_{n}^{2}} = \sqrt{{(\frac{2 a s}{m})}^{2} + {(\frac{2 a s^{2}}{m R})}^{2}} = \frac{2 a s}{m} \sqrt{1 + {(\frac{s}{R})}^{2}}$
Hence the sought force, $F = m w = 2 a s \sqrt{1 + {(\frac{s}{R})}^{2}}$

Suggest Corrections

Similar questions

Q. The kinetic energy

K

of a particle moving along a circle of radius

R

depends on the distance covered as

K = a s^{2},

where

a

is a constant. The force acting on the particle is

Q. The kinetic energy

T

of a particle moving along a circle of radius

R

depends on the distance covered as

T = a s^{2}

. The force acting on the particle is:

The kinetic energy $k$ of a particle moving along a circle of radius $R$ depends on the distance covered $s$ as $k = a s^{2}$ where $a$ is a constant. The force acting on the particle is

Q. The kinetic energy of a particle moving along a circle of radius

R

depends on distance

(s)

K = A s^{2}

where

A

is a constant.

The tangential force acting on the particle is

Q. The Kinetic energy ‘K’ of a particle moving along a circle of radius R depends on the distance covered ‘S’ as

K = a s^{2}

. The force acting on the particle is

Join BYJU'S Learning Program

The kinetic energy of a particle moving along a circle of radius R depends on the distance covered s as T=as2, where a is a constant. Find the force acting on the particle as a function of s.

We have T=12mv2=as2 or v2=2as2m (1)Differentiating equation (1) w.r.t. time2vwt=4asmv or, wt=2asm (2)Hence net acceleration of the particlew=√w2t+w2n=√(2asm)2+(2as2mR)2=2asm√1+(sR)2Hence the sought force, F=mw=2as√1+(sR)2

The kinetic energy of a particle moving along a circle of radius $R$ depends on the distance covered $s$ as $T = a s^{2}$ , where $a$ is a constant. Find the force acting on the particle as a function of $s$ .