Angular velocity and angular momentum of rod

[zhuyilun]

Homework Statement

A uniform rod of mass 2.90×10^−2 kg and length 0.410m rotates in a horizontal plane about a fixed axis through its center and perpendicular to the rod. Two small rings, each with mass 0.160 kg, are mounted so that they can slide along the rod. They are initially held by catches at positions a distance 4.90×10−2^2 on each side from the center of the rod, and the system is rotating at an angular velocity 28.0 rad/s. Without otherwise changing the system, the catches are released, and the rings slide outward along the rod and fly off at the ends. 1)find the angular velocity when the rings reach the end of the rod 2)fly off the rod

Homework Equations

The Attempt at a Solution

angular momentum= I*w
so i thought the initial angular momentum is (2.9*10^2*(0.41/2)^2+0.16*2*(4.9*10^-2)^2)*28
since angular momentum is conserved due to the absence of net torque, the final momentum should be equal to the initial which is (2.9*10^2*(0.41/2)^2+0.16*2*(0.41/2)^2)*w, w is what i need for the first part. but this is wrong

 

[tiny-tim]

Hi zhuyilun!

(have an omega: ω and try using the X² tag just above the Reply box )

… so i thought the initial angular momentum is (2.9*10^2*(0.41/2)^2+0.16*2*(4.9*10^-2)^2)*28

No, you've used mL² for the moment of inertia of a rod of length 2L about its centre …

look up a list of common moments of inertia (eg http://en.wikipedia.org/wiki/List_of_moments_of_inertia" ), and learn them!

 

[kerimek]

, what am i missing?

Your approach is correct, but there are a few errors in your calculations. First, the mass of the rings should be included in the moment of inertia calculation, so it should be (2.9*10^-2*(0.41/2)^2+0.16*2*(0.41/2)^2) instead of (2.9*10^2*(0.41/2)^2+0.16*2*(0.41/2)^2). Additionally, the distance from the center of the rod to the rings should be 4.9*10^-2 instead of 4.9*10^-2^2. This will give you the correct initial angular momentum.

For part 1, you can use the conservation of angular momentum equation to solve for the final angular velocity. This will give you an equation of the form I*w = I*w', where I is the initial moment of inertia and w' is the final angular velocity. Solving for w' will give you the angular velocity when the rings reach the end of the rod.

For part 2, you can use the fact that the rings fly off the rod when they reach the ends, which means that they have a tangential velocity equal to the end of the rod. You can use this velocity to calculate the angular velocity at which the rings fly off the rod using the equation v = r*w, where v is the tangential velocity, r is the distance from the center of rotation to the rings, and w is the angular velocity.

Remember to use consistent units in your calculations and be careful with any conversions that need to be done.