A uniform rod hangs vertically on a frictionless horizontal axle passing through its center. The rod is 35cm long and 85g. A 15g ball of clay travelling horizontally at 2.5cm/s hits and sticks to the bottom of the rod. What is the maximum angle the rod makes with the vertical after the collision?
To find the maximum angle the rod makes with the vertical after the collision, we can apply the principle of conservation of angular momentum.
Angular momentum is the product of moment of inertia and angular velocity. In this scenario, since the axle is frictionless and the rod rotates about its center, the moment of inertia remains constant.
Before the collision, the rod is hanging vertically, so it is not rotating. Hence, the initial angular momentum is zero.
After the collision, the clay sticks to the bottom of the rod, creating an off-center of mass. This off-center of mass causes the rod to rotate.
Let's denote the mass of the rod as M and the length of the rod as L. The mass of the clay is m and its velocity is v.
The final angular momentum of the system is the sum of the angular momentum of the rod and the angular momentum of the clay. Since the rod is rotating, its angular momentum is given by:
L_rod = I * ω,
where I is the moment of inertia of the rod and ω is the angular velocity.
The moment of inertia of a rod rotating about its center is given by:
I = (1/12) * M * L^2.
The angular velocity ω can be calculated from the linear velocity v of the clay at the point of impact. Since the clay is stuck to the bottom of the rod, the effective length of the rod increases by the radius of the clay sphere.
Let's denote the radius of the clay as r. After the collision, the effective length is L_eff = L + r.
Therefore, the angular velocity ω is given by:
ω = v / L_eff.
Now we can calculate the final angular momentum of the system:
L_final = (1/12) * M * L^2 * (v / L_eff) + m * v * (L + r),
where m is the mass of the clay.
Since angular momentum is conserved, we can equate the initial and final angular momentum:
0 = (1/12) * M * L^2 * (v / L_eff) + m * v * (L + r).
Now we can solve for the maximum angle θ by setting up the equation:
tan(θ) = (L_eff * ω) / g,
where g is the acceleration due to gravity.
Substituting the values and solving the equation will give you the maximum angle the rod makes with the vertical after the collision.
Note: The numerical values for M, L, m, r, v, and g would need to be provided in order to calculate the exact value of the maximum angle.