An object of mass,m,is placed in an arc of a circle of radius,r.The coefficient of kinetic friction between the object and the surface of the arc is $\mu$_k.What is the work done by friction in going from P₁ and P₂? A)-$\mu$_kmg r (cos($\theta$₂)-cos($\theta$₁)) B)-$\mu$_kmg r (sin($\theta$₂)-sin($\theta$₁)) C)-$\mu$_kmg r cos($\theta$₂)sin$\theta$₁) D)-$\mu$_kmg r sin($\theta$₂)cos($\theta$₁) E)none of the above

Question

Quizplus · Accepted Answer

The work done by friction can be calculated using the formula: 
Wfriction = Frictional Force x distance moved in the opposite direction of the force 
In this case, the distance moved is the arc length of the circle between the initial and final points. This can be calculated using the formula s = rθ, where θ is the angle between the two points as seen from the center of the circle. 
The frictional force is given by Ffriction = μmg, where μ is the coefficient of kinetic friction, m is the mass of the object, and g is the acceleration due to gravity. 
Using these formulas, we can calculate the work done by friction as: 
Wfriction = μmg r (θ2 - θ1) 
Taking the negative sign since the frictional force acts in the opposite direction of the displacement, we get: 
Wfriction = -μmg r (θ2 - θ1)

Substituting the values given in the options, we see that option B is the only one that matches this equation.

An Object of Mass,m,is Placed in an Arc of a Circle