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 7.1 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 find the angular velocity of the straight rod, we can use the principle of conservation of angular momentum. According to this principle, the total angular momentum of an isolated system remains constant if no external torques act on it.
In this case, the system consists of only the rod. Initially, when the rod is bent upward, it has an angular velocity of 7.1 rad/s about an axis perpendicular to its left end and parallel to the upward half. Let's call this axis the "original axis."
When the rod suddenly assumes its straight shape, the total angular momentum of the system must remain the same. Therefore, the angular momentum about the original axis is conserved.
The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. Since the rod is uniform and thin, its moment of inertia can be expressed as (1/3) * m * L^2, where m is the mass of the rod and L is its length.
Let's assume that the mass of the rod is M, and its length is 2L (since it's bent at the midpoint). So, the moment of inertia initially is (1/3) * M * (2L)^2.
When the rod straightens, its length is L, and we need to find the final angular velocity. Let's call this angular velocity w.
Since the total angular momentum is conserved, we can set up the following equation:
(1/3) * M * (2L)^2 * 7.1 = (1/3) * M * L^2 * w
Simplifying the equation, we find:
(2^2) * 7.1 = w
w = 4 * 7.1
w = 28.4 rad/s
Therefore, the angular velocity of the straight rod is 28.4 rad/s.
To solve this problem, we can use the principle of conservation of angular momentum. When the bent rod straightens, the angular momentum of the system will be conserved.
Initially, the bent rod is rotating only about its center of mass with an angular velocity of 7.1 rad/s. Let's assume the mass of the rod is m, the length of the rod is L, and the distance of the center of mass from the left end is d.
The initial angular momentum can be calculated as:
L_initial = I_initial * ω_initial
Where L_initial is the initial angular momentum, I_initial is the moment of inertia of the bent rod, and ω_initial is the initial angular velocity.
The moment of inertia of the bent rod can be calculated as:
I_initial = (1/12) * m * L^2 + m * d^2
Where (1/12) * m * L^2 is the moment of inertia of a uniform rod rotating about its center of mass and m * d^2 is the moment of inertia of a point mass rotating about an axis perpendicular to it.
As the rod straightens, the moment of inertia changes. The new moment of inertia for a straight rod can be calculated as:
I_final = (1/3) * m * L^2
Where (1/3) * m * L^2 is the moment of inertia of a uniform rod rotating about its center of mass.
Since the angular momentum is conserved, we can equate the initial and final angular momentum:
L_initial = I_final * ω_final
Plugging in the values, we get:
I_initial * ω_initial = I_final * ω_final
Substituting the moment of inertia values, we have:
[(1/12) * m * L^2 + m * d^2] * ω_initial = (1/3) * m * L^2 * ω_final
Simplifying the equation, we get:
(1/12) * L^2 * ω_initial + d^2 * ω_initial = (1/3) * L^2 * ω_final
Rearranging the equation for ω_final, we have:
ω_final = (1/3) * [((1/12) * L^2 * ω_initial + d^2 * ω_initial) / L^2]
Simplifying further:
ω_final = (1/3) * [(1/12) * ω_initial + d^2 * ω_initial / L^2]
Substituting the given values, we can calculate the final angular velocity of the straight rod.