A uniform rod of mass 2.0 kg and length 8.4 m is free to rotate about one end (see the following figure). If the rod is released from rest at an angle of 60° with respect to the horizontal, what is the speed (in m/s) of the tip of the rod as it passes the horizontal position?

Question

A uniform rod of mass 2.0 kg and length 8.4 m is free to rotate about one end (see the following figure). If the rod is released from rest at an angle of 60° with respect to the horizontal, what is the speed (in m/s) of the tip of the rod as it passes the horizontal position?

Answer 1

To find the speed of the tip of the rod as it passes the horizontal position, we can use the principles of rotational motion.

Let's break down the problem into smaller steps:

Step 1: Identify the known variables and parameters:
- Mass of the rod (m) = 2.0 kg
- Length of the rod (L) = 8.4 m
- Release angle (θ) = 60°

Step 2: Determine the moment of inertia of the rod:
The moment of inertia of a uniform rod rotating about one end can be given by the formula:
I = (1/3) * m * L^2

Substituting the known values:
I = (1/3) * 2.0 kg * (8.4 m)^2

Step 3: Calculate the gravitational torque:
When the rod is released, the gravitational force acting at the center of mass of the rod creates a torque. The torque can be given by:
τ = I * α

In this case, as the rod is released from rest, it is at an angle of 60°. At the horizontal position, the angle becomes 0°. The change in angle is 60° - 0° = 60°.

So, the angular acceleration (α) can be calculated using the equation:
α = change in angular velocity / change in time

Since the rod is released from rest, the initial angular velocity is 0. Therefore, the change in angular velocity is the final angular velocity (ω) at the horizontal position.

Step 4: Calculate the final angular velocity (ω):
The final angular velocity (ω) can be calculated using the equation:
ω^2 = ω0^2 + 2αθ

Since the initial angular velocity (ω0) is 0, the equation simplifies to:
ω = sqrt(2 * α * θ)

Substituting the known values:
ω = sqrt(2 * α * 60°)

Step 5: Calculate the speed of the tip of the rod as it passes the horizontal position:
The speed of the tip can be given by:
v = ω * L

Substituting the known values:
v = ω * 8.4 m

Now, you can plug in the known values into the equations and solve for the speed (v) of the tip of the rod as it passes the horizontal position.