Two 0.85Mtrs -long pendulums hang side by side. The masses of the pendulum bobs are 50g and 70g . The lighter bob is pulled aside until its string is horizontal and is then released from rest. It swings down and collides elastically with the other bob at the bottom of its arc.
Part A
To what height does the lighter bob rebound?
To determine the height to which the lighter bob rebounds after the collision, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system (consisting of both pendulums) remains constant throughout the motion.
Let's break down the problem into steps to find the answer:
Step 1: Calculate the potential energy of the lighter bob at its initial position.
The potential energy (PE) of an object is given by the formula PE = mgh, where m represents the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height above some reference point. In this case, we can consider the initial position of the lighter bob when the string is horizontal as the reference point.
Using the given information:
Mass of the lighter bob (m1) = 50g = 0.05kg
Height above the reference point (h1) = 0 (since the reference point is the initial position)
PE1 = m1 * g * h1
= 0.05kg * 9.8 m/s^2 * 0
= 0 Joules
Step 2: Calculate the kinetic energy of the lighter bob just before the collision.
The kinetic energy (KE) of an object is given by the formula KE = (1/2) * m * v^2, where m represents the mass of the object and v is the velocity of the object.
Since the lighter bob is released from rest, its initial velocity (v1) will be zero.
KE1 = (1/2) * m1 * v1^2
= (1/2) * 0.05kg * 0^2
= 0 Joules
Step 3: Calculate the potential energy of the lighter bob immediately after the collision.
The potential energy (PE) of an object is given by the same formula as in Step 1.
Since the lighter bob rebounds and reaches its maximum height after the collision, the potential energy in this case will be at its maximum. Let's denote this final height as h2.
Using the given information:
Height above the reference point (h2) = maximum height reached by the lighter bob after rebounding
PE2 = m1 * g * h2
Step 4: Calculate the kinetic energy of the lighter bob immediately after the collision.
Since this is an elastic collision, the total mechanical energy of the system will be conserved. Therefore, any energy lost by the lighter bob during the collision will be transferred as kinetic energy to the heavier bob.
Using the principle of conservation of mechanical energy:
Initial mechanical energy = Final mechanical energy
KE1 + PE1 = KE2 + PE2
Since we have determined KE1 and PE1 to be zero in steps 2 and 1 respectively:
0 + 0 = KE2 + PE2
Therefore, KE2 + PE2 = 0
Since we know KE2 = (1/2) * m1 * v2^2, and we have already calculated PE2 using Step 3, we can solve for v2 (the velocity of the lighter bob immediately after the collision).
Finally, we can calculate the height to which the lighter bob rebounds, h2, using the equation:
v2^2 = v_initial^2 - 2g(h2 - h_initial)
where v_initial is the initial velocity (0 in this case), g is the acceleration due to gravity, and h_initial is the initial height (0 in this case).
By rearranging the equation, we can solve for h2:
h2 = (v_initial^2 - v2^2) / (2g)
Substituting the known values, we can find the height to which the lighter bob rebounds.