The sides of a cone make an angle ϕ with the vertical. A small mass m is placed on the inside of the cone and?
The sides of a cone make an angle ϕ with the vertical. A small mass m is placed on the inside of the cone and the cone, with its point down, is revolved at a frequency f (revolutions per second) about its symmetry axis.If the coefficient of static friction is μs, at what positions on the cone can the mass be placed without sliding on the cone?(Give the maximum and minimum distances, r, from the axis).
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at the minimum distance, the friction force points upslope and prevents the mass from sliding down the cone.
normal force Fn = mgsinφ + m(2πf)²r*cosφ
and the friction force (upslope) is Ff = µs*Fn.
Another upslope force comes from the centripetal acceleration,
Fc = m(2πf)²r*sinφ
The downslope force comes from the weight: Fg = mgsinφ
At the threshold, Fg = Fc + Ff
Substitute and solve for r.
At the maximum distance, the friction force points downslope, so
Fg + Ff = Fc