A projectile of mass m moves to the right with a speed vi. The projectile strikes and sticks to the end of a stationary rod of mass M, of length d, that is pivoted about a frictionless axle through its center.
(a) Find the angular speed of the system right after the collision. (Use v_i for vi, m, M, and d as appropriate in your equation.)
(b) Determine the fractional loss in mechanical energy due to the collision. (Use v_i for vi, m, M, and d as appropriate in your equation.)
fractional loss of energy
I got part a correct and it's (12mv_i^2)/((m+M)d^2)
For part b is it just the final kinetic energy over the initial kinetic energy? If so, why do I keep getting the same answer as part a?
for part b, your final KE is rotational KE which is now 1/2 I w^2 where I consists of the rod mass distributed, and the mass of the initial stuff at some rotational distance. You cant get the same answer as a), as the units are different. THe units in a) are sec^-1, and in b) should be unitless.
so would it be (Iw^2)/(mv^2)?
Yes, depending on what you put in for final I, w, and the intial v.
actually, since b) asked for fractional loss, it would be that fraction subtracted from 1.
What did you put in for I
To solve part b, you need to calculate the initial and final mechanical energies of the system. The mechanical energy of a system is the sum of its kinetic energy and potential energy.
First, let's calculate the initial kinetic energy of the system. The projectile has kinetic energy due to its linear motion, given by 1/2 * m * vi^2. The rod is stationary, so it has no initial kinetic energy.
Next, let's calculate the final kinetic energy of the system. Since the projectile sticks to the end of the rod, the entire system moves as one unit. The system now has rotational motion, so its kinetic energy is given by 1/2 * (m + M) * v^2, where v is the final velocity of the system.
To find v, we can use the conservation of linear momentum. Before the collision, the total linear momentum is given by m * vi, and after the collision, the total linear momentum is given by (m + M) * v. Since momentum is conserved, we have:
m * vi = (m + M) * v
Solving for v, we get:
v = (m * vi) / (m + M)
Now, we can plug this value of v into the equation for the final kinetic energy to calculate it.
Finally, the fractional loss in mechanical energy is given by the difference between the initial and final mechanical energies, divided by the initial mechanical energy. Mathematically, it is expressed as:
Fractional Loss = (Initial Mechanical Energy - Final Mechanical Energy) / Initial Mechanical Energy
Remember that mechanical energy is the sum of kinetic and potential energy. Since the problem assumes there is no change in gravitational potential energy, the change in potential energy is negligible and can be ignored.
Thus, by calculating the initial and final kinetic energies and using the equation above, you can determine the fractional loss in mechanical energy due to the collision.