A particle of mass m moves in the x-y plane, along a symmetric, frictionless, concave curve. The trajectory length along this curve is denoted by s. A uniform gravitational field acts in the negative y-direction. The particle is to move along this curve with simple harmonic motion with angular frequency w, and the angle of displacement of m from the y-axis is not limited to small values. Find the locus of the curve of the motion in terms of the given parameters.
Hint: Let q be the angle between the normal to the curve and the vertical direction. Find x and y of the curve as a function of q.
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When a particle of mass m oscillates about its
equilibrium position on a frictionless circular track, simple harmonic motion occurs for
small oscillations. If we define the position of the particle by the arc length s,
then 
If the displacement is not small, restriction to the circular orbit will not result in
simple harmonic motion. Suppose a track of non-circular shape is used to make the motion
simple harmonic even for large oscillations. Using 
find the parametric equations of the
trajectory x(q) and y(q).
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A time-dependent force F(t) of finite duration acts
on an one-dimensional simple harmonic oscillator of mass m and resonant frequency w. Show that the equation of motion can be written as 
where 
Next show that the energy
transferred to the oscillator in the limit tое can be
written as 
where the oscillator
is initially at rest and equilibrium. Finally, for the force F=F0 for 0<t<t0
and 0 for all other times, where F0 is a constant, find the
duration t0 of the force that transfers maximum energy to the
oscillator.
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