Two identical spheres of mass are initially at rest on a smooth horizontal surface

PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials

Problem

Two identical spheres of mass

m

are initially at rest on a smooth horizontal surface. Each sphere is acted upon by a horizontal force for the same duration

0\le t\le T

, but the time–dependence of the force differs for the two spheres. (a) For sphere A the force is

F_{A}(t)=F_{0}\sin\left(\dfrac{\pi t}{T}\right)

, where

F_{0}

is its maximum value at

t=\dfrac{T}{2}

. Show that the total impulse delivered to sphere A during the interval

0\le t\le T

\dfrac{2F_{0}T}{\pi}

. (b) Sphere B experiences a constant horizontal force

F_{B}

acting over the same time interval. If the two spheres acquire identical final momenta, determine

F_{B}

in terms of

F_{0}

. (c) Compare the maximum force acting on sphere A with that on sphere B and comment on the practical significance of your result in situations where peak force must be minimised.

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