In the figure, a small 0.125 kg block slides down a frictionless surface through height h = 0.958 m and then sticks to a uniform vertical rod of mass M = 0.250 kg and length d = 2.32 m. The rod pivots about point O through angle θ before momentarily stopping. Find θ.
To find the angle θ, we need to consider the conservation of mechanical energy and angular momentum.
First, let's determine the initial gravitational potential energy of the small block before it slides down the surface. The gravitational potential energy is given by the formula:
PE_initial = m * g * h
where m is the mass of the block (0.125 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height through which the block slides (0.958 m).
PE_initial = 0.125 kg * 9.8 m/s^2 * 0.958 m
Next, when the block sticks to the rod, it transfers its potential energy into rotational kinetic energy. The total mechanical energy after the block sticks to the rod is equal to the sum of the initial gravitational potential energy and the final rotational kinetic energy of the rod.
Let's calculate the final rotational kinetic energy of the rod. The rotational kinetic energy is given by the formula:
KE_final = (1/2) * I * ω^2
where I is the moment of inertia of the rod about the pivot point O, and ω is the angular velocity of the rod.
The moment of inertia for a uniform rod pivoting about one end is given by the formula:
I = (1/3) * M * d^2
where M is the mass of the rod (0.250 kg) and d is the length of the rod (2.32 m).
I = (1/3) * 0.250 kg * (2.32 m)^2
Now, let's consider angular momentum. The angular momentum is given by the formula:
L = I * ω
where L is the angular momentum, and I and ω are the same as before.
Now, since there is no external torque acting on the system, the angular momentum is conserved before and after the block sticks to the rod. Therefore, we can equate the initial angular momentum of the block to the final angular momentum of the rod. The initial angular momentum is given by:
L_initial = m * r_initial * v_initial
where r_initial is the distance of the small block from the pivot point O initially, and v_initial is the velocity of the small block just before it sticks to the rod.
Finally, solve the conservation of mechanical energy and angular momentum equations to find θ. This involves setting the initial gravitational potential energy equal to the final rotational kinetic energy of the rod, and setting the initial angular momentum equal to the final angular momentum.
Once you have the value of θ, you can calculate its numerical value.