Static Equilibrium Condition - General Physics - Solved Past Paper, Exams of Physics

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A uniform rectangular beam of length L= 5 m and mass M=40 kg is supported but not attached to the two posts which are length D=3 m apart. A child of mass W=20 kg starts walking along the beam.

a) Assuming infinitely rigid posts, how close can the child get to the right end of the beam without it falling over? [Hint: The upward force exerted by the left on the beam cannot be negative – this is the limiting condition on how far the child can be to the right. Set up the static equilibrium condition with the pivot about the left end of the beam.]

A child of mass M is on a swing whose seat and rope have negligible mass, and the rope has length L. Suppose the child can be treated as a point mass, and the rope makes an angle as a function of time θ(t) with respect to the vertical. a) Neglecting any friction in the system and when θ(t) is much smaller than unity and when there is nobody pushing the child, the differential equation satisfied by θ(t) is

What is B in terms of g and L?

b) Solve the equation

for θ(t) if

are given as initial conditions (where B and C are positive numbers).

c) For the motion given in part b), what is the child’s maximum kinetic energy?

d) Suppose an adult can push the child in the direction of child’s motion just when the child passes through θ=0. To achieve maximum height for the child, approximately how many times should the adult push the child during a time period Q which is much longer than the natural oscillation period of the swing? [Hint: Think resonance.]