Auniform, thin rod of length h and mass m is held vertically with its lower end resting on a frictionless horizontal surface. the rod is then released to fall freely. (a) what is the speed of its center of mass just before it hits the horizontal surface? (use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.) vcm =

To solve this exercise it is necessary to apply the concepts related to the conservation of rotational, kinetic and potential energy, as well as the concepts of moments of inertia in this type of bodies.

By definition we know that,

Where,

KE = Kinetic energy

U = Potential energy

Let us start by defining that the center of mass of the body is located at a distance h / 2 from the bar and that the moment of inertia of a bar is defined by

Where M means the mass and h the height, then,

There is not potential energy at the beginning and also there is not Kinetic energy at the end then

Replacing inertia values,

Re-arrange for v, we have

Note that the value for the angular velocity ()we replace with the equivalent in tangential velocity, which is , where v is the velocity and R the radius, at this case h/2

Therefore the center of mass hat a velocity equal to