A thin stick of mass M = 3.1 kg and length L = 1.6 m is hinged at the top. A piece of clay, mass m = 0.7 kg and velocity V = 4.0 m/s hits the stick a distance x = 1.20 m from the hinge and sticks to it.
1. What is the angular velocity of the stick immediately after the collision?
2. What is the ratio of the final mechanical energy to the initial mechanical energy?
angular momentum around pivot is the same before and after
before
angular momentum = m v r = .7*4*1.2
= 3.36 kg m^2/s
after let w = omega = ang velocity about pivot
angular momentum = I w
I = (1/3)(3.1)(1.6^2) + .7(1.2^2)
= 2.645 + 1.008 = 3.65
so
I w = 3.36
3.65 w = 3.36
w = .921 radians/second
part 2
before
(1/2) m v^2 = (1/2).7(16) = 5.6 Joules
after
(1/2) I w^2 = (1/2)(3.65)(.921^2) = 1.55 Joules
1.55/5.6 = .276
To solve these questions, we need to apply the principle of conservation of angular momentum and conservation of mechanical energy.
1. Angular Momentum Conservation:
The angular momentum of the system before the collision is equal to the angular momentum immediately after the collision.
Given:
Mass of the stick, M = 3.1 kg
Length of the stick, L = 1.6 m
Mass of the clay piece, m = 0.7 kg
Distance of impact from the hinge, x = 1.20 m
Velocity of the clay piece, V = 4.0 m/s
Angular momentum before the collision:
L_initial = (moment of inertia of the stick) * (angular velocity of the stick before collision)
= (1/3) * M * L^2 * ω_initial -------- (1)
Angular momentum immediately after the collision:
L_final = (moment of inertia of the stick with clay) * (angular velocity of the stick after collision)
= (1/3) * (M + m) * L^2 * ω_final -------- (2)
Since no external torque is acting on the system, angular momentum is conserved. Therefore, L_initial = L_final.
Substituting the values from equations (1) and (2) and solving for ω_final:
(1/3) * M * L^2 * ω_initial = (1/3) * (M + m) * L^2 * ω_final
Simplifying and solving for ω_final:
ω_final = (M * ω_initial) / (M + m) -------- (3)
Substituting the given values into equation (3) will give us the angular velocity of the stick immediately after the collision.
2. Mechanical Energy Conservation:
The mechanical energy of the system remains constant when no external forces are doing work on it. Therefore, we can compare the initial and final mechanical energies.
Initial mechanical energy (E_initial) = (kinetic energy of the stick) + (kinetic energy of the clay piece)
Final mechanical energy (E_final) = (kinetic energy of the stick with clay)
Using the formulas for kinetic energy:
E_initial = (1/2) * M * (angular velocity of the stick before collision)^2 + (1/2) * m * V^2
E_final = (1/2) * (M + m) * (angular velocity of the stick after collision)^2
The ratio of final mechanical energy to initial mechanical energy:
E_final / E_initial
Substituting the given values into the above equation will give us the required ratio.
Note: To obtain the numerical solution, substitute the given values into the respective equations and perform the calculations.