A 1.0-kg steel ball and a 2.0-m cord of negligible mass makeup a simple pendulum that can pivot without friction about thepoint O, as in the picture below. This pendulum isreleased from rest in a horizontal position, and when the ball isat its lowest point it strikes a 1.0-kg block sitting at rest on ashelf. Assume that the collision is perfectly elastic andthat the coefficient of kinetic friction between the block andshelf is 0.10.
(a) What is the velocity of the lbock just after impact?
(b) How far does the block slide before coming to rest(assuming that the shelf is long enough)?
To answer these questions, we need to apply the laws of conservation of momentum and conservation of energy.
(a) To find the velocity of the block just after impact, we can use the principle of conservation of momentum. Since the collision is perfectly elastic, the total momentum before the collision will be equal to the total momentum after the collision.
Let's assume the velocity of the steel ball just before the collision is v and the velocity of the block just after the collision is V.
The total momentum before the collision is given by:
Initial momentum = mass of the ball * velocity of the ball
Since the block is initially at rest, its initial momentum is zero.
The total momentum after the collision is given by:
Final momentum = mass of the ball * velocity of the ball + mass of the block * velocity of the block
Using the principle of conservation of momentum, we can set up the equation:
mass of the ball * velocity of the ball = mass of the ball * v + mass of the block * V
Since the mass of the ball and the block are both 1.0 kg, we can simplify the equation to:
v = V
Therefore, the velocity of the block just after impact is equal to the velocity of the steel ball just before the collision.
(b) To find how far the block slides before coming to rest, we can use the principle of conservation of energy. The initial energy of the system is the potential energy of the steel ball at its highest point, which is equal to its mass times the acceleration due to gravity (9.8 m/s^2) times its height (which is equal to the length of the pendulum, 2.0 m).
The final energy of the system is the energy dissipated due to friction between the block and the shelf, which is equal to the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance over which it acts.
To calculate the force of friction, we can use the equation:
Force of friction = coefficient of friction * normal force
The normal force is the force exerted by the shelf on the block and is equal to the weight of the block, which is equal to its mass times the acceleration due to gravity.
Using the equation for work done by friction:
Work done by friction = force of friction * distance
We can set up the equation for conservation of energy:
Initial energy = Final energy
mass of the ball * gravitational acceleration * height = force of friction * distance
Substituting the expressions for the force of friction and the height:
mass of the ball * gravitational acceleration * height = (coefficient of friction * mass of the block * gravitational acceleration) * distance
Simplifying the equation:
height = coefficient of friction * mass of the block * distance
Rearranging for distance:
distance = height / (coefficient of friction * mass of the block)
Substituting the given values (height = 2.0 m, coefficient of friction = 0.10, mass of the block = 1.0 kg):
distance = 2.0 m / (0.10 * 1.0 kg)
Therefore, the block slides a distance of 20 m before coming to rest.