How Do You Calculate the Velocity of a Corner of a Falling Tilted Plate?

[rosie2985]

Homework Statement

A square plate is tilted on one of its corners as shown below. If the corner does not slip, and the plate is allowed to fall, determine the velocity of the moving corner before it hits the ground. The moving corner is point P, which is 45 degrees above the horizontal.
Mass=1.8kg
Length of each side=.1m
Angle_{center of mass}=80 degrees
Angle_{point P}=45 degrees

Homework Equations

Mg(Y_final-&_initial)=.5M omega²

The Attempt at a Solution

I don't understand how to find delta Y in order to find potential energy. The center of mass is .071 m.

 

[Andrew Mason]

Welcome to PF. I am not clear on the plate's position. You will either have to post the picture or do a better job of describing how the plate is sitting before it starts to fall. Thanks.

AM

 

[kerimek]

I would approach this problem by first understanding the concept of rotational motion and its associated equations. In this case, we are dealing with a situation where a square plate is tilted and allowed to fall, resulting in rotational motion.

To find the velocity of the moving corner before it hits the ground, we need to use the equation for conservation of energy in rotational motion. This equation states that the change in potential energy (due to gravity) is equal to the change in kinetic energy (due to rotation).

In this case, we can calculate the change in potential energy by finding the difference in height between the initial and final positions of the center of mass. This can be done by using trigonometry to find the vertical component of the distance between the initial and final positions of the center of mass.

Next, we can use the equation for rotational kinetic energy, which states that the kinetic energy of a rotating object is equal to 1/2 times the moment of inertia (I) times the angular velocity (ω) squared. The moment of inertia for a square plate can be calculated using its mass and dimensions.

By equating the change in potential energy to the change in kinetic energy, we can solve for the angular velocity (ω). Once we have the angular velocity, we can use it to find the linear velocity of the moving corner (point P) using the relationship v = ωr, where r is the distance from the center of mass to point P.

In summary, to find the velocity of the moving corner before it hits the ground, we need to use the equations for conservation of energy in rotational motion, rotational kinetic energy, and the relationship between angular and linear velocity. By solving for the angular velocity, we can then find the linear velocity of the moving corner.