Conservation laws

Problem:

A simple way to measure the speed of a bullet is with a ballistic pendulum.  As illustrated, this consists of a wooden block of mass M into which the bullet is shot horizontally.  The block is suspended from cables of length l, and the impact of the bullet causes it to swing through a maximum angle Φ, as shown.  The initial speed of the bullet is v and its mass is m.
(a)  What is the speed of the block V immediately after the bullet comes to rest inside the wooden block?  (Assume that this happens quickly.)
(b)  Find an expression for the speed v of the bullet in terms of the quantities that can be easily measured m, M, g, l, and Φ.

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Solution:

Problem:

An plane, inclined at θ = 20o, touches a wall as shown in the picture.  You drop a small, perfectly elastic ball from a height h = 1.5 m onto the onto the plane.  The ball falls from rest.  You do not move your hand.  At what distance d from the wall do you have to drop it so that it bounces back into your hand?

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Solution:

Problem:

Two wedges are placed mirror symmetrically so that the tip of one touches the tip of another as shown in the figure below.  The surface of each wedge is at angle θ = 30 degrees relative to the ground.  A small elastic ball is dropped from height h = 1 m with zero initial velocity.  How far from the tips of the two wedges (x) must the small ball be dropped, so that after bouncing from the two wedges it will reach the same height from where it was dropped?  Neglect any friction from air and consider the bouncing of the ball to be completely elastic.

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Solution:

Problem:

A wheel of radius b is rolling along a muddy road with a speed v.  Particles of mud attached to the wheel are being continuously thrown off from all points of the wheel.  If v2 > 2bg, where g is the acceleration of gravity, find the maximum height above the road attained by the mud, H = H(b,v,g).

Solution:

Problem:

In one dimension, a particle is acted upon by an attractive force F = −kx3.
(a)  Show that the period for the motion of this particle is inversely proportional to the amplitude.
(b)  By contrast show that the period for the motion of a particle subjected to the force F = −kx is independent of the amplitude.

Solution:

Problem:

A particle of mass m is subjected to a force whose potential energy is U(x) = ax2 – bx3,
with a and b constants and a > 0.
(a)  Find the force.
(b)  Assume that the particle starts at the origin with velocity of magnitude v0.  Show that if
v0 < vC, where vC is a certain critical velocity, the particle will be confined in a region near the origin.  Find vC.

Solution:

Problem:

A ball is dropped down an elevator shaft.  The elevator has an upward speed V.  The instant the ball is dropped, the top of the elevator is below it by a distance h.  How high will the ball rebound above the point from which it was dropped?

Solution:

More problems

Energy conservation and circular motion in a uniform gravitational field

Work and energy

Gravity