In the figure, a thin uniform rod of mass m = 3.05 kg and length L = 1.11 m rotates freely about a horizontal axis A that is perpendicular to the rod and passes through a point at a distance d = 0.201 m from the end of the rod. The kinetic energy of the rod as it passes through the vertical position is K = 8.63 J. At what angle £c (in ¢X) will the rod momentarily stop in its upward swing?
I don't really know how to solve the inertia for this one. I got my w value from my r, as L-d and K. Now I was confused is I value.
I calculated my inertia as m(L^2)/12 +m(L-d)^2
To solve for the angle at which the rod momentarily stops in its upward swing, we need to find the moment of inertia of the rod, knowing its mass, length, and the distance of the axis of rotation from the end of the rod.
The moment of inertia of a thin rod rotating about an axis perpendicular to it and passing through one end is given by the formula: I = (1/3) * m * L^2
In this case, however, the axis of rotation is at a distance d from the end of the rod. To account for this, we need to apply the parallel axis theorem.
The parallel axis theorem states that the moment of inertia about an axis parallel to and at a distance 'a' from an axis passing through the center of mass is given by: I' = I + m * a^2
In this case, the axis passing through the center of mass is located at distance L/2 from each end of the rod. Thus, a = L/2 - d.
Substituting this value into the parallel axis theorem, we can calculate the moment of inertia of the rod about axis A:
I' = (1/3) * m * L^2 + m * (L/2 - d)^2
I' = (1/3) * m * L^2 + m * (L/2)^2 - 2m * d * (L/2) + m * d^2
I' = (1/3) * m * L^2 + (1/4) * m * L^2 - m * d * L + m * d^2
I' = (7/12) * m * L^2 - m * d * L + m * d^2
Now we can use the law of conservation of energy to find the angle at which the rod momentarily stops. The initial kinetic energy K is equal to the potential energy at its highest position:
K = (1/2) * I' * w^2
8.63 J = (1/2) * ((7/12) * m * L^2 - m * d * L + m * d^2) * w^2
Solve this equation for w^2, and then solve for w using the given information about the rod's length and distance from the axis. Finally, calculate the angle from the angular velocity using the formula:
w = v / r
where v is the linear velocity at the end of the rod and r is the distance from the end of the rod.