A steel ball of mass m1 = 1.1 kg and a cord of length of L = 2.3 m of negligible mass make up a simple pendulum that can pivot without friction about the point O, as in the figure below. This pendulum is released from rest in a horizontal position, and when the ball is at its lowest point it strikes a block of mass m2 = 1.1 kg sitting at rest on a shelf. Assume that the collision is perfectly elastic and that the coefficient of kinetic friction between the block and shelf is 0.10.
(a) What is the velocity of the block just after impact?
?m/s
(b) How far does the block slide before coming to rest (assuming that the shelf is long enough)?
?m
To solve this problem, we can apply the principle of conservation of linear momentum and energy conservation.
(a) First, let's find the velocity of the ball just before the impact by using the principle of conservation of energy. At its lowest point, the ball has converted all its gravitational potential energy to kinetic energy:
m1 * g * L = (1/2) * m1 * v^2
where g is the acceleration due to gravity and v is the velocity of the ball just before the impact.
Simplifying the equation:
v^2 = 2 * g * L
v = √(2 * 9.8 * 2.3)
v ≈ 10.96 m/s
Now, let's use the principle of conservation of linear momentum to find the velocity of the block just after the impact. Since the collision is perfectly elastic, the total momentum before the collision is equal to the total momentum after the collision:
m1 * v1 = m1 * v1' + m2 * v2'
where v1 is the velocity of the ball just before the impact, v1' is the velocity of the ball just after the impact, m2 is the mass of the block, and v2' is the velocity of the block just after the impact.
Plugging in the values:
(1.1 kg) * (10.96 m/s) = (1.1 kg) * v1' + (1.1 kg) * v2'
10.96 = v1' + v2'
Since the ball and block are initially at rest, v1' = 0. Therefore,
10.96 = v2'
So, the velocity of the block just after impact is approximately 10.96 m/s.
(b) To find how far the block slides before coming to rest, we need to consider the work done by the friction force. The work done by friction is equal to the change in kinetic energy of the block:
friction force * distance = (1/2) * m2 * v2'^2
The friction force can be calculated using the coefficient of kinetic friction:
friction force = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the block:
normal force = m2 * g
Substituting these values:
coefficient of kinetic friction * m2 * g * distance = (1/2) * m2 * v2'^2
(m2 * g * distance) = (1/2) * v2'^2
Substituting the value of v2' (10.96 m/s):
(m2 * g * distance) = (1/2) * (10.96)^2
Simplifying the equation:
distance = (10.96)^2 / (2 * g)
Substituting the value of g (9.8 m/s^2):
distance ≈ 6.06 m
So, the block slides approximately 6.06 meters before coming to rest.
To solve this problem, we need to break it down into two parts: determining the velocity of the block just after impact and calculating the distance the block slides before coming to rest.
(a) To find the velocity of the block just after impact, we can use the principle of conservation of momentum. Before the collision, the pendulum is at rest, so the initial momentum is zero. After the collision, the pendulum ball and the block move together as a single system.
1. Calculate the initial total momentum:
- The pendulum ball has no initial velocity, so its momentum is zero.
- The block has no initial velocity, so its momentum is also zero.
2. Calculate the final total momentum:
- Since the collision is perfectly elastic, kinetic energy is conserved. Thus, the final total momentum is also zero.
3. Apply the principle of conservation of momentum:
Initial total momentum = Final total momentum
0 = m1 * v1 + (m1 + m2) * v2
where v1 is the velocity of the pendulum ball just before the collision, v2 is the velocity of the block just after the collision, m1 is the mass of the pendulum ball, and m2 is the mass of the block.
4. Solve the equation for v2:
v2 = - (m1 * v1) / (m1 + m2)
(b) To calculate the distance the block slides before coming to rest, we need to consider the work done by the force of kinetic friction.
1. Determine the work done by the friction force:
The work done by friction is equal to the friction force multiplied by the distance traveled. The friction force is given by the equation:
friction force = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the block, which can be calculated as:
normal force = m2 * g
where g is the acceleration due to gravity.
2. Calculate the distance traveled by the block:
The work done by friction is equal to the decrease in kinetic energy of the block. Thus, we can express the work done by friction as:
work done by friction = change in kinetic energy of the block
force of friction * distance = 0.5 * m2 * v2^2
Rearrange the equation to solve for the distance:
distance = (0.5 * m2 * v2^2) / (coefficient of kinetic friction * m2 * g)
Now, substitute the given values for mass (m1 and m2), coefficient of kinetic friction, and acceleration due to gravity (g), and use the calculated value of v2 to find the answers to the two parts of the question.