Deriving equation of motion for Overhead Crane

[cs23]

Homework Statement

1)Derive equation of motion for trolley. 2)Derive the equation of motion for the load using balance of torques. 3)Then relate the force applied to the angular displacement of the load.

Coefficient of friction between trolley and rail is negligible. For the moment of inertia of the load, use I(load) = 0.1 (kg m2).
b(load)=1.75Ns/m
m(load)=0.3kg
m(trolley)=18kg
L=2m

I've attached the picture with the variables

Homework Equations

The equations are attached in the document

The Attempt at a Solution

My solution is in the attachment.

Could someone check it over

 

[KataKoniK]

and make sure everything looks good?

Thank you for your post. I have reviewed your solution and it seems to be correct. Here is a breakdown of the steps:

1) To derive the equation of motion for the trolley, we first need to draw a free body diagram and apply Newton's second law. We have the force of gravity acting downwards, the normal force from the rail acting upwards, and the force of tension from the rope acting towards the left. Using the equation ΣF=ma, we can solve for the acceleration of the trolley.

2) To derive the equation of motion for the load, we need to consider the balance of torques. The only torque acting on the load is the force of tension from the rope, which causes the load to rotate clockwise. We also have the moment of inertia of the load, which can be calculated using the given information. The equation for torque is τ=Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. We can solve for the angular acceleration and then use the relationship between angular and linear motion to determine the linear acceleration of the load.

3) Finally, we can relate the force applied to the angular displacement of the load using the equation τ=Iα. Since we know the force of tension and the moment of inertia, we can solve for the angular acceleration and then use the relationship between angular and linear motion to determine the linear acceleration of the load. This allows us to see how the force applied affects the displacement of the load.

Overall, your solution is correct and well-explained. Keep up the good work!