A weather vane initially at rest has a moment of inertiaof 0.100 kg-m2about its axis of rotation. A 50.0 g pieceof clay is thrown at the vane and sticks to it at a point20.0 cm from the axis. The initial velocity of the clay is 15.0 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, we can apply the principle of conservation of angular momentum. The initial angular momentum of the system (weather vane + clay) should be equal to the final angular momentum.
Angular momentum (L) is given by the formula:
L = I * ω
Where:
L = angular momentum
I = moment of inertia
ω (omega) = angular velocity
The initial angular momentum of the system is given by the clay's angular momentum when it sticks to the vane. The clay's initial angular momentum (L_initial) can be calculated using the formula:
L_initial = m * r * v
Where:
m = mass of the clay
r = distance of the clay from the axis
v = initial velocity of the clay
Let's substitute the given values into the above formula:
m = 50.0 g = 0.050 kg (converting grams to kilograms)
r = 20.0 cm = 0.20 m (converting centimeters to meters)
v = 15.0 m/s
L_initial = 0.050 kg * 0.20 m * 15.0 m/s
L_initial = 0.150 kg·m²/s
Now, since angular momentum is conserved, the final angular momentum (L_final) should be equal to the initial angular momentum.
L_final = L_initial
L_final = I * ω
Substituting the values:
I * ω = 0.150 kg·m²/s
I = 0.100 kg·m² (moment of inertia, given)
Now we can rearrange the equation to solve for ω:
ω = L_final / I
Substituting the values:
ω = 0.150 kg·m²/s / 0.100 kg·m²
ω = 1.5 rad/s
Therefore, the angular velocity of the weather vane just after it is struck is 1.5 rad/s.