Angular velocity of a falling rod?

• spursfan2110
In summary, the conversation discusses a uniform rod of mass 0.62 kg that is pivoted about a horizontal, frictionless pin at the end of a thin extension. The rod is initially at an angle of 48 degrees with the horizontal and is released from rest at the same angle. The question is to find the angular speed of the rod at the instant it is in a horizontal position, given the acceleration of gravity and the moment of inertia of the rod. Using the parallel axis theorem, the correct moment of inertia is found to be (1/12)mL2 + md2. Substituting this value into the equation for conservation of energy, the angular speed is solved to be 5.397339822 rad/s.
spursfan2110
A uniform rod of mass 0.62 kg is 6 m long.
The rod is pivoted about a horizontal, fric-
tionless pin at the end of a thin extension (of
negligible mass) a distance 6 m from the cen-
ter of mass of the rod. Initially the rod makes
an angle of 48◦ with the horizontal. The rod
is released from rest at an angle of 48◦ with
the horizontal, as shown in the figure.
What is the angular speed of the rod at
the instant the rod is in a horizontal position?
The acceleration of gravity is 9.8m/s2 and the
moment of inertia of the rod about its center
of mass is I = (1/12)mL2

U = mgh
KE = .5Iw2

The Attempt at a Solution

So, I set U = KE

mgh = .5Iw2

Then I plugged in the moment of inertia

mgh = .5(1/12)(mass of rod)(length of rod)2w2

Cancelled out the m and solved for the height of the center of mass

g(6sin(48degrees)) = .5(1/12)mL2w2

and finally, I plugged in values and solved.

(9.8)(6sin(48 degrees)) = .5(1/12)(6)2w2

And solved for w, which I found to equal 5.397339822.

This is wrong, but I have no idea where my error is. Can you guys help me spot it??

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parallel axis theorem, since the system is not rotating about its center of mass

Yeah, you have to use Steiner's theorem to work out the inertia at that point.

So if I read my stuff right, instead of inserting (1/12)mL2 for momentum, I would insert (1/12)mL2 + md2?

What is the definition of angular velocity?

Angular velocity is a measure of the rate of change of angular displacement of an object with respect to time. It is represented by the symbol ω (omega) and is measured in radians per second.

How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the change in time. It can also be calculated by multiplying the radius of rotation by the linear velocity.

What is the relationship between angular velocity and linear velocity?

Angular velocity and linear velocity are directly related. The linear velocity of a point on a rotating object is equal to the angular velocity multiplied by the distance from the point to the axis of rotation.

Does the angular velocity of a falling rod change?

Yes, the angular velocity of a falling rod will change as it falls due to the influence of gravity and air resistance. As the rod falls, its angular velocity will increase until it reaches terminal velocity.

How does the length of the falling rod affect its angular velocity?

The length of the falling rod does not affect its angular velocity. However, it can affect the rod's moment of inertia, which is a factor in determining its angular velocity. A longer rod will have a larger moment of inertia and therefore a lower angular velocity compared to a shorter rod with the same mass and shape.

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