A weather vane initially at rest has a moment of inertia of 0.115 kg · m2 about its axis of rotation. A 44.5 g piece of clay is thrown at the vane and sticks to it at a point 10.5 cm from the axis. The initial velocity of the clay is 18.5 m/s, directed perpendicular to the vane. Find the angular velocity of the weather vane just after it is struck.
The moment of inertia of weather vane with piece of clay is
I = I(vane) +mR^2 = 0.115+44.5•10^-3•(0.105)^2=
= 1.2•10^-2 kg • m2.
The torque due to the clay hitting is M=F•R
F•t=Δp=m•v, therefore F=m•v/t.
Then M=F•R=m•v•R/t.
From Newton’s 2 Law for rotation M=I• ε.
m•v•R/t =I•ε.
m•v•R/I= ε• t = ω
ω = m•v•R/I ==44.5•10^-3•18.5•0.105/1.2•10^-2= 7.21 rad•s
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.
The angular momentum of the system before the clay strikes the vane is given by:
L_initial = I * ω_initial
Where:
L_initial is the initial angular momentum,
I is the moment of inertia of the weather vane, and
ω_initial is the initial angular velocity of the weather vane.
Since the weather vane is initially at rest, the initial angular velocity (ω_initial) is zero.
After the clay sticks to the vane, the total angular momentum of the system is conserved. We can express the final angular momentum as:
L_final = I * ω_final
Where:
L_final is the final angular momentum after the clay sticks to the vane, and
ω_final is the angular velocity of the weather vane just after it is struck.
According to the conservation of angular momentum, the initial and final angular momentum should be the same. Hence,
L_initial = L_final
I * ω_initial = I * ω_final
0 = I * ω_final
Since the moment of inertia (I) is nonzero, the final angular velocity (ω_final) must be zero.
Therefore, the angular velocity of the weather vane just after it is struck by the clay is zero (0).
To find the angular velocity of the weather vane just after the clay is struck, we can use the principle of conservation of angular momentum.
Angular momentum is defined as the product of moment of inertia (I) and angular velocity (ω). According to the conservation of angular momentum, the total angular momentum before the clay is struck is equal to the total angular momentum after the clay is struck.
Let's calculate the initial angular momentum before the clay is struck:
Initial angular momentum of the weather vane = Initial moment of inertia × Initial angular velocity
The weather vane is initially at rest, so the initial angular velocity is 0. Therefore, the initial angular momentum is also 0.
Now let's calculate the final angular momentum after the clay is struck:
Final angular momentum of the weather vane = Final moment of inertia × Final angular velocity
The final moment of inertia of the weather vane can be calculated by adding the moment of inertia of the vane itself and the moment of inertia of the clay attached to it:
Final moment of inertia = Moment of inertia of the vane + Moment of inertia of the clay
The moment of inertia of the vane is given as 0.115 kg·m².
To calculate the moment of inertia of the clay, we need to use the parallel axis theorem. According to the parallel axis theorem, the moment of inertia of an object about any axis parallel to and a distance "d" from the object's center of mass is equal to the sum of the moment of inertia about the center of mass and the product of the object's mass and the square of the distance "d".
Moment of inertia of the clay = Moment of inertia about the center of mass + (mass of the clay × distance of clay from the axis)²
The distance of the clay from the axis is given as 10.5 cm, which is 0.105 m. The mass of the clay is given as 44.5 g, which is 0.0445 kg.
Now we can calculate the moment of inertia of the clay and the final moment of inertia of the weather vane:
Moment of inertia of the clay = 0.0445 kg × (0.105 m)²
Final moment of inertia = 0.115 kg·m² + (0.0445 kg × (0.105 m)²)
Finally, we can use the conservation of angular momentum to find the final angular velocity:
Initial angular momentum = Final angular momentum
0 = (0.115 kg·m²) × 0 + (0.0445 kg × (0.105 m)²) × Final angular velocity
Solving for the Final angular velocity:
Final angular velocity = 0 / (0.0445 kg × (0.105 m)²)
Final angular velocity = 0 m/s
Therefore, the angular velocity of the weather vane just after the clay is struck is 0 m/s.