To solve for the tension in the cord to prevent the body from rotating about the pin, we will follow the steps provided:
a) Identify coordinate systems, constraints, and the degrees of freedom of the system:
- Coordinate system: We will use a Cartesian coordinate system, where the vertical axis is the z-axis.
- Constraints: The constraint in this system is that the body is pinned to a rotating shaft.
- Degrees of freedom: Since the body is pinned, it can rotate only about the vertical axis. Therefore, it has one degree of freedom.
b) Find expressions for the absolute acceleration of the center of mass (a_G) and the angular velocity (ω˙) of the body:
- The absolute acceleration of the center of mass, a_G, will be zero since the center of mass is not moving.
- The angular velocity, ω, will be equal to the constant angular speed, ω, of the rotating shaft.
c) Draw a free body diagram of the forces and moments acting on the body:
- In the free body diagram, we will have the weight acting downward at the center of mass, and the tension in the cord acting upward at the point of connection.
d) Write the force balance (∑ F = m a_G) to get 3 scalar equations:
- Since a_G is zero, the force balance equation will be:
∑ F = 0
T - mg = 0
T = mg
e) Write the moment balance (∑ M = dt dH_0) to get 3 scalar equations:
- The moment balance equation will be:
∑ M = 0
τ - Izz * ω˙ = 0
Where τ represents the torque applied by the tension in the cord, and Izz represents the moment of inertia of the body about the vertical axis.
f) Solve for the tension in the rope, T:
From the force balance equation (T = mg) and the moment balance equation (τ - Izz * ω˙ = 0), we can see that the tension in the rope, T, is equal to the weight of the body, mg.
Therefore, the tension in the cord required to prevent the body from rotating about the pin is equal to the weight of the body, mg.