A piece of putty of mass m = 0.75 kg and velocity v = 2.5 m/s moves on a horizontal frictionless surface. It collides with and sticks to a rod of mass M = 2 kg and length L = 0.9 m which pivots about a fixed vertical axis at the opposite end of the rod as shown. What fration of the initial kinetic energy of the putty is lost in this collision?
KElost/KEinitial =
0.4712
Well, let's see. The fraction of the initial kinetic energy lost in this collision can be given by the equation:
KElost/KEinitial = (KEinitial - KEfinal) / KEinitial
Now, to calculate this, we need to know the final kinetic energy of the system. Since the putty sticks to the rod, they move together as one object. The final velocity of the system can be found using the principle of conservation of linear momentum:
m1v1 + m2v2 = (m1 + m2)Vf
Where m1 and v1 are the mass and velocity of the putty, m2 is the mass of the rod, and Vf is the final velocity of the system.
In this case, m1 = 0.75 kg, v1 = 2.5 m/s, m2 = 2 kg, and we need to find Vf.
0.75 kg * 2.5 m/s + 2 kg * 0 = (0.75 kg + 2 kg) * Vf
1.875 kg*m/s = 2.75 kg * Vf
Vf = 1.875 kg*m/s / 2.75 kg
Vf ≈ 0.682 m/s
Now, we can calculate the initial kinetic energy (KEinitial) and the final kinetic energy (KEfinal) using the equations:
KEinitial = 0.5 * m1 * v1^2
KEfinal = 0.5 * (m1 + m2) * Vf^2
KEinitial = 0.5 * 0.75 kg * (2.5 m/s)^2
KEfinal = 0.5 * (0.75 kg + 2 kg) * (0.682 m/s)^2
KEinitial ≈ 1.47 J
KEfinal ≈ 0.725 J
Finally, we can calculate the fraction of the initial kinetic energy lost:
KElost/KEinitial = (KEinitial - KEfinal) / KEinitial
KElost/KEinitial ≈ (1.47 J - 0.725 J) / 1.47 J
KElost/KEinitial ≈ 0.507
So, approximately 50.7% of the initial kinetic energy of the putty is lost in this collision.
To solve this problem, we need to consider the conservation of angular momentum and the conservation of mechanical energy.
1. Conservation of angular momentum:
The initial angular momentum of the system (putty + rod) is zero since the putty is moving towards the rod along the axis of rotation. After the collision, the putty sticks to the rod, so the final angular momentum is also zero.
2. Conservation of mechanical energy:
The initial kinetic energy of the putty is given by KE_initial = 0.5 * m * v^2, where m is the mass of the putty and v is its velocity. The final kinetic energy is zero as the putty sticks to the rod.
Therefore, the fraction of the initial kinetic energy lost in this collision is given by:
KE_lost/KE_initial = (KE_initial - KE_final) / KE_initial
Since KE_final is zero, we can simplify the equation to:
KE_lost/KE_initial = KE_initial / KE_initial
Substituting the values given:
KE_lost/KE_initial = (0.5 * 0.75 * (2.5)^2) / (0.5 * 0.75 * (2.5)^2) = 1
Therefore, 100% of the initial kinetic energy of the putty is lost in this collision.
To determine the fraction of the initial kinetic energy of the putty that is lost in this collision, we need to calculate the initial kinetic energy and the final kinetic energy of the system.
The initial kinetic energy (KE_initial) of the system can be calculated using the formula:
KE_initial = 0.5 * m * v^2
Where:
m = mass of the putty (0.75 kg)
v = velocity of the putty before the collision (2.5 m/s)
Substituting the values, we get:
KE_initial = 0.5 * 0.75 kg * (2.5 m/s)^2
KE_initial = 0.9375 Joules
Now, let's calculate the final kinetic energy (KE_final) of the system after the collision. Since the putty sticks to the rod, they will move together as a single object.
The system's center of mass (CM) will have a velocity (V_CM) after the collision.
Using the principle of conservation of momentum, we can write:
(m * v) + (M * 0) = (m + M) * V_CM
The term (M * 0) represents the initial velocity of the rod which is zero.
Rearranging the equation and solving for V_CM, we get:
V_CM = (m * v) / (m + M)
Substituting the given values, we have:
V_CM = (0.75 kg * 2.5 m/s) / (0.75 kg + 2 kg)
V_CM = 1.25 / 2.75 m/s
V_CM ≈ 0.455 m/s
Now, we can calculate the final kinetic energy (KE_final) using the formula:
KE_final = 0.5 * (m + M) * V_CM^2
Substituting the values, we get:
KE_final = 0.5 * (0.75 kg + 2 kg) * (0.455 m/s)^2
KE_final = 0.5 * 2.75 kg * (0.455 m/s)^2
KE_final ≈ 0.351 Joules
The fraction of the initial kinetic energy lost in the collision can be calculated as:
KE_lost/KE_initial = (KE_initial - KE_final) / KE_initial
Substituting the values, we have:
(0.9375 J - 0.351 J) / 0.9375 J ≈ 0.627
So, the fraction of the initial kinetic energy of the putty that is lost in this collision is approximately 0.627.