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A particle of mass m moves along the qua...

A particle of mass m moves along the quarter section of the circular parth whose centre is at the origin. The radius of the circular path is a. A force `vecF=yhati-x hatj` N acts on the particle, where x, y denote the coordinates of the position of the particle. Calculate the work done by this force in taking the particle from point A(a, 0) to point B(0, a) along the circular path.

Text Solution

Verified by Experts

The correct Answer is:
`-(pia^(2))/(2)`
Work done by force `F ,`
`w=int vec(F).dvec(r)=int (yhat(i)-xhat(j)).(dxhat(i)+dyhat(j))`
`:. x^(2)+y^(2)=a^(2) :. xdx +y dy=0 `
`rArrW=int(y((-ydy)/(x))-xdy)`
`-int((x^(2)+y^(2)))/(x)dy`
`=-int_(0)^(a)(a^(2))/(sqrt(a^(2)-y^(2)))dy=-(pia^(2))/(2)`
Aternate Method
It can be observed that the force is tangent to the curve at each point and the magnitude is constant. The direction of force is opposite to the direction of motion of the particle.
`:. ` work done =(force) `xx` (distance)
`=-sqrt(x^(2)+y^(2))(pia)/(2)=-axx(pia)/(2)=-(pia^(2))/(2)J`
Ans. `w=-(pia^(2))/(2)J`
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