Challenging Problems in Mechanics to sharpen your mind

by Pravin Sharma

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

Ans: Velocity=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 V_L=(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 un-stretched rope is L. Force constant of the rope is K and gravitational field strength is g. Mass of rope is negligible, air resistance 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

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