MEDIUM
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IMPORTANT
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A ballerina begins a tour jeté with angular speed ωi and a rotational inertia consisting of two parts: Ileg=1.36 kg m2 for her leg extended outward at an angle of θ=90° to her body and Itrunk =0.620 kg m2 for the rest of her body (primarily her trunk). Near her maximum height, she holds both legs at an angle of θ=30° to her body and has an angular speed of ωf. Assuming that Itrunk has not changed, what is the ratio of ωfωi?

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Important Questions on Rolling, Torque, and Angular Momentum

HARD
JEE Main
IMPORTANT

Two 2.00 kg balls are attached to the ends of a thin rod of length 50.0 cm and negligible mass. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its center. With the rod initially horizontal (Figure), a 50.0 g wad of wet putty drops onto one of the balls. Hitting it with a speed of 3.00 m s-1 and then sticking to it. (a) What is the angular speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the system after the collision and that of the putty wad just before? (c) Through what angle will the system rotate before it momentarily stops?

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HARD
JEE Main
IMPORTANT

In the figure shown below, a small 50 g block slides down a frictionless surface through height h=15 cm and then sticks to a uniform rod of mass100 g  and length 35 cm. The rod pivots about point O through an angle θ before momentarily stopping. Find θ.

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MEDIUM
JEE Main
IMPORTANT

The figure is an overhead view of a thin uniform rod of length 0.600 m and mass M  rotating horizontally at 80.0 rad s-1 counter clockwise about an axis through its center. A particle of mass M/3.00 and traveling horizontally at speed 40.0 m s-1 hits the rod and sticks. The particle's path is perpendicular to the rod at the instant of the hit, at a distance d from the rod's center. (a) At what value of d are the rod and the particle stationary after the hit? (b) In which direction does the rod and the particle rotate if d is greater than this value?

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EASY
JEE Main
IMPORTANT
A top spins at 25 rev/s about an axis that makes an angle of 30o with the vertical. The mass of the top is 0.50 kg, its rotational inertia about its central axis is  5.0×10-4 kg·m2, and its center of mass is 4.2 cm from the pivot point. If the spin is clockwise from an overhead view, what are the (a) precession rate and (b) direction of the precession as viewed from overhead?
EASY
JEE Main
IMPORTANT
A certain gyroscope consists of a uniform disk with a 50 cm radius mounted at the center of an axle that is 11 cm long and of negligible mass. The axle is horizontal and supported at one end. If the spin rate is 1000 rev/min, what is the precession rate?