A war-wolf, or trebuchet, is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fling pumpkins and pianos. A simple trebuchet is shown in the figure. Model it as a stiff rod of negligible mass 3.00 m long and joining particles of mass 74.0 kg and 0.280 kg at its ends. It can turn on a frictionless horizontal axle perpendicular to the rod and 25.0 cm from the particle of larger mass. The rod is released from rest in a horizontal orientation. Find the maximum (a) angular speed.
To find the maximum angular speed of the trebuchet, we can use the principle of conservation of angular momentum. This principle states that the total angular momentum of a system remains constant unless acted upon by an external torque.
In this case, the trebuchet starts from rest and rotates freely about a frictionless horizontal axle. The trebuchet is released from rest in a horizontal orientation, meaning it rotates in only one plane.
The formula for angular momentum is given by:
L = I * ω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
To find the moment of inertia of the system, we need to consider the moments of inertia of the two particles on either end of the rod.
The moment of inertia of a point mass rotating about an axis perpendicular to its motion is given by:
I = m * r^2
Where m is the mass of the particle and r is the distance from the axis of rotation.
For the particle of mass 74.0 kg at the end of the rod:
I1 = (74.0 kg) * (0.25 m)^2 = 4.625 kg*m^2
For the particle of mass 0.280 kg at the other end of the rod:
I2 = (0.280 kg) * (2.75 m)^2 = 2.557 kg*m^2
The total moment of inertia of the system is the sum of the individual moments of inertia:
I_total = I1 + I2 = 4.625 kg*m^2 + 2.557 kg*m^2 = 7.182 kg*m^2
Since the trebuchet is released from rest, the initial angular momentum is zero:
L_initial = 0
At the maximum angular speed, the trebuchet reaches its maximum angular momentum:
L_max = I_total * ω_max
Since L_initial = L_max, we have:
0 = 7.182 kg*m^2 * ω_max
Solving this equation for ω_max, we find that the maximum angular speed is 0 rad/s.
Therefore, the maximum angular speed of the trebuchet is 0 rad/s.