A thin non-uniform rod of length L=2.00 m and mass M=9.00 kg is free to pivot about an axis at one end. The CM of the rod is at a distance d=1.30 m from that end as illustrated below. The rod's moment of inertia about an axis through its CM and parallel to the rotation axis is ICM=2.57 kg m2.

The rod is released at rest from an angle θ=33.0°. Calculate the speed of the center of mass when the rod is horizontal.

A thin uniform rod (length = 1.4 m, mass = 2.5 kg) is pivoted about a horizontal, frictionless pin through one end of the rod. (The moment of inertia of the rod about this axis is 1/3 ML2.) The rod is released when it makes an angle of 45° with the horizontal. What is the angular acceleration of the rod at the instant it is released?

To calculate the speed of the center of mass when the rod is horizontal, we can use the principle of conservation of mechanical energy.

First, we need to determine the potential energy of the rod when it is released at an angle of 33.0°. The potential energy can be calculated using the formula:

PE = m * g * h

Where:
m = mass of the rod
g = acceleration due to gravity
h = height of the center of mass (distance from the rotation axis)

Given:
m = 9.00 kg
g = 9.8 m/s^2 (approximate value of acceleration due to gravity)
h = 1.30 m

Substituting the values into the formula, we get:

PE = 9.00 kg * 9.8 m/s^2 * 1.30 m
PE = 117.06 J

Next, we need to determine the kinetic energy of the rod when it is horizontal. The kinetic energy can be calculated using the formula:

KE = (1/2) * I * ω^2,

Where:
I = moment of inertia about the axis of rotation
ω = angular speed of the rod (in radians per second)

Given:
I = 2.57 kg m^2 (moment of inertia about the center of mass axis)
θ = 33.0° (angle when released)

First, we need to convert the angle from degrees to radians. To do this, we use the conversion factor:

θ (in radians) = θ (in degrees) * (π/180)

Substituting the values, we get:

θ (in radians) = 33.0° * (π/180)
θ (in radians) ≈ 0.575958653 radians

Now, we can calculate the angular speed using the formula:

ω = √((2 * PE) / I)

Substituting the known values, we get:

ω = √((2 * 117.06 J) / 2.57 kg m^2)
ω ≈ 10.998 radians/s

Finally, the speed of the center of mass can be calculated using the formula:

v = ω * r

Where:
v = speed of the center of mass
r = distance from the rotation axis to the center of mass

Given:
r = 1.30 m (distance from the rotation axis to the center of mass)

Substituting the values, we get:

v = (10.998 radians/s) * 1.30 m
v ≈ 14.297 m/s

Therefore, the speed of the center of mass when the rod is horizontal is approximately 14.297 m/s.