Challenging Problems in Mechanics to sharpen your mind

1.A particle of mass M is acted upon by a force that has a intial magnitude F0 when t=0 and decrease at a uniform rate until when t=t1,its magnitude is zero.It is moving accross the X-axis.Find the velocity and coordinate of the particle when t=t2 assuming that t2 > t1. Assume xt=0=0 and (dx/dt)t=0=0

Ans
Velcity=F0t1/2M
Displacement=(F0t1/2M)(t2-t1/3)


2.A wheel of radius r rolls without slip along the x-axis. with constant speed. v0.Find out the motion(velocity and acceleration ) of the point A on the rim of the wheel which starts from the origin O.Take Y axis as perpendicular at X axis at origin

Ans

dx/dt=v0[1-cos(v0t/r)]
dy/dt=v0sin(v0t/r)

d2x/dt2=(v02/r)sin(v0t/r)
d2y/dt2=(v02/r)cos(v0t/r)

3.A body moves under the action of a constant force F through fluid that opposes the motion with a force proportional to the square of the velocity that is Ax2.Show that the limiting velocity is VL=(F/A)1/2.

4.A Bungee Jumper is attached to one end of a long elastic rope.The other end of the elastic rope is fixed to a high bridge.The Jumper steps off the bridge and falls from rest towards the river below..he does not hit the river below.The mass of the jumper is M and length of unstretched rope is L.Force constant of the rope is K. and gravitional field strength is g.Mass of rope is negligible ,air resitance is negligible.
1.Find out the distance y dropped by the jumper before coming instantaneously to rest for the first time
2.Maximum speed attained by the jumper during this drop
3.The time taken during the drop before coming to rest for the first time


Answer
y=[KL+mg+√(2mgKL+m2g2)]/k
v=√(2gL+mg2/k)
t=√(2L/g) + √(m/k)tan-1{-√(2KL/mg)}

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