A thin, uniform rod is hinged at its midpoint. To begin with, one half of the rod is bent upward and is perpendicular to the other half. This bent object is rotating at an angular velocity of 6.8 rad/s about an axis that is perpendicular to the left end of the rod and parallel to the rod’s upward half (see the drawing). Without the aid of external torques, the rod suddenly assumes its straight shape. What is the angular velocity of the straight rod?
The angular velocity of the straight rod is 0 rad/s.
To solve this problem, we can use the concept of conservation of angular momentum. According to this concept, the initial angular momentum of the system will be conserved when the rod straightens out, assuming there are no external torques acting on it.
Let's define the variables:
- Initial angular velocity of the bent rod = ω1 = 6.8 rad/s
- Angular velocity of the straight rod = ω2 (which we need to find)
- Length of the rod = L
Step 1: Determine the initial angular momentum
The angular momentum of an object is given by the equation: L = Iω, where:
- L is the angular momentum
- I is the moment of inertia
- ω is the angular velocity
The moment of inertia of a thin uniform rod rotating about its midpoint is given by the equation: I = (1/12)mL², where:
- m is the mass of the rod
- L is the length of the rod
Since the rod is thin and uniform, we can assume that the mass is distributed evenly throughout its length.
The initial angular momentum (L1) can be calculated as follows:
L1 = I1 * ω1
Step 2: Determine the final angular momentum
When the rod straightens out, the moment of inertia changes to that of a straight rod rotating about one end. The moment of inertia of a straight rod rotating about one end is given by the equation: I = (1/3)mL².
The final angular momentum (L2) can be calculated as follows:
L2 = I2 * ω2
I2 = (1/3)mL²
Step 3: Apply the conservation of angular momentum
According to the conservation of angular momentum, L1 = L2, so:
I1 * ω1 = I2 * ω2
(1/12)mL² * ω1 = (1/3)mL² * ω2
Simplify the equation by canceling mL²:
(1/12) * ω1 = (1/3) * ω2
Step 4: Solve for ω2
Multiply both sides of the equation by 12 to get:
ω2 = 12/3 * ω1
ω2 = 4 * ω1
Substitute the value of ω1:
ω2 = 4 * 6.8 rad/s
ω2 ≈ 27.2 rad/s
So, the angular velocity of the straight rod is approximately 27.2 rad/s.
To determine the angular velocity of the straight rod, we need to apply the principle of conservation of angular momentum.
Angular momentum (L) is a property of rotating objects and is defined as the product of the moment of inertia (I) and the angular velocity (ω).
In equation form: L = I * ω
Since the rod is initially rotating, it has an initial angular momentum (L_initial) given by L_initial = (1/2) * M * (L/2)^2 * ω_initial.
Here, M represents the mass of the rod, L represents the length of the rod, and ω_initial represents the initial angular velocity.
When the rod becomes straight, its moment of inertia changes since it transforms from a bent shape to a straight shape. The new moment of inertia (I_new) of a straight rod rotating about its center is given by I_new = M * (L^2 / 12).
Since there are no external torques acting on the system, the total angular momentum before and after the transformation remains the same. Therefore, we can equate the initial and final angular momentum.
L_initial = L_final
(1/2) * M * (L/2)^2 * ω_initial = M * (L^2 / 12) * ω_final
Now, we can solve for ω_final, which is the angular velocity of the straight rod.
First, let's simplify the equation:
(1/4) * L^2 * ω_initial = (L^2 / 12) * ω_final
Now, we can cancel out the L^2 terms:
(1/4) * ω_initial = (1/12) * ω_final
Next, let's isolate ω_final:
ω_final = (4/12) * ω_initial = (1/3) * ω_initial
Finally, we can substitute the initial angular velocity ω_initial = 6.8 rad/s into the equation:
ω_final = (1/3) * 6.8 rad/s = 2.27 rad/s
Therefore, the angular velocity of the straight rod is approximately 2.27 rad/s.