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 just after it is struck by the clay, we can use the principle of conservation of angular momentum.
Angular momentum (L) is defined as the product of moment of inertia (I) and angular velocity (ω). Mathematically, it can be expressed as: L = I * ω
Before the clay hits the vane, it is at rest, so the initial angular momentum (L_initial) is zero.
After the clay sticks to the vane, the final angular momentum (L_final) is given by the sum of the angular momentum of the weather vane and the angular momentum of the clay.
The angular momentum of the clay is calculated by multiplying its moment of inertia with its angular velocity. Since the clay sticks to the vane, the angular velocity of the clay becomes the same as the vane's angular velocity.
Given:
Moment of inertia of the vane, I = 0.119 kg · m^2
Mass of the clay, m = 48.0 g = 0.048 kg (converted to kg)
Distance of the clay from the axis, r = 21.0 cm = 0.21 m (converted to meters)
Initial velocity of the clay, v = 23.0 m/s
First, let's calculate the moment of inertia of the clay. The moment of inertia of a point mass about an axis passing through its center of mass is given by the formula I = m * r^2.
Substituting the values:
Moment of inertia of the clay, I_clay = (0.048 kg) * (0.21 m)^2
Next, we can calculate the angular momentum of the clay, L_clay:
L_clay = I_clay * ω_clay
Since the clay sticks to the vane, the angular velocity of the clay is the same as the vane's angular velocity. So, L_clay = I * ω, where I is the moment of inertia of the vane (given).
Now, we can set up the equation for conservation of angular momentum as:
L_initial = L_final
0 = (I_vane * ω_vane) + (I_clay * ω_clay)
Since L_initial is zero, we can rearrange the equation to solve for the final angular velocity of the vane (ω_vane):
ω_vane = -(I_clay * ω_clay) / I_vane
Substituting the given values, we can calculate the angular velocity of the vane.