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.]
b) Suppose the left end of the beam is attached to the left post although it can still freely
pivot about it. The right post is not attached as in part a). If the left post is infinitely
rigid while the Young’s modulus for the right post which is of length s=10 m is Y=1010
N/m2 and its cross sectional area is A=1/2 m2, how much does its length differ between
the situation when the child is not present and when the child is at the rightmost edge 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.]