A weather vane initially at rest has a moment of inertia of 0.119 kg · m2 about its axis of rotation. A 48.0 g piece of clay is thrown at the vane and sticks to it at a point 21.0 cm from the axis. The initial velocity of the clay is 23.0 m/s, directed perpendicular to the vane. Find the angular velocity of the weather vane just after it is struck.
To find the angular velocity of the weather vane after it is struck, we can use the principle of conservation of angular momentum. Before the clay is thrown, the weather vane is at rest, so its initial angular momentum is zero.
The angular momentum of an object rotating about an axis is given by the equation:
L = I * ω
Where:
L is the angular momentum,
I is the moment of inertia of the object,
ω is the angular velocity.
After the clay sticks to the vane, the angular momentum of the system (weather vane + clay) is conserved. Therefore, the total angular momentum before and after the clay is thrown is equal.
Let's calculate the initial angular momentum of the clay. The moment of inertia of the weather vane is given as 0.119 kg·m^2. The clay sticks to the weather vane at a point 21.0 cm (or 0.21 m) from the axis. The initial velocity of the clay is 23.0 m/s.
The moment of inertia of the combined system (weather vane + clay) is given by:
I_total = I_vane + m * r^2
Where:
I_total is the total moment of inertia,
I_vane is the moment of inertia of the weather vane,
m is the mass of the clay,
r is the distance between the clay and the axis.
Substituting the given values:
I_total = 0.119 kg·m^2 + 0.048 kg * (0.21 m)^2
Now, we can use the conservation of angular momentum to find the angular velocity of the weather vane after the collision. Since the initial angular momentum is zero, we have:
L_initial = L_final
0 = I_initial * ω_initial = I_final * ω_final
Solving for ω_final:
ω_final = (I_initial * ω_initial) / I_final
Substituting the known values:
ω_final = (0 * ω_initial) / I_total
Since the initial angular velocity is perpendicular to the vane, it becomes ω_initial = v_initial / r, where v_initial is the initial velocity of the clay and r is the distance between the clay and the axis.
Substituting the known values:
ω_final = (0 * (23.0 m/s / 0.21 m)) / I_total
Therefore, the angular velocity of the weather vane just after it is struck is zero.