Chapter 11: Rolling, Torque, and Angular Momentum

In Figure, a small $\mathbf{50}\mathbf{g}$ block slides down a frictionless surface through height $\mathbf{h}\mathbf{=}\mathbf{}\mathbf{20}\mathbf{}\mathbf{cm}$and then sticks to a uniform rod of mass $\mathbf{100}\mathbf{}\mathbf{g}$and length$\mathbf{40}\mathbf{}\mathbf{cm}$ . The rod pivots about point Othrough angle $\mathbf{\theta }$before momentarily stopping. Find$\mathbf{\theta }$

Figure is an overhead view of a thin uniform rod of length $\mathbf{0}\mathbf{.600}\mathbf{}\mathbf{m}$and mass Mrotating horizontally at $\mathbf{80.0}\mathbf{}\mathbf{rad}\mathbf{/}\mathbf{s}$counter clock-wise about an axis through its centre. A particle of mass $\mathbf{M}\mathbf{/}\mathbf{3.00}$and travelling horizontally at speed $\mathbf{40.0}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$hits the rod and sticks. The particle’s path is perpendicular to the rod at the instant of the hit, at a distance $\mathit{d}$from the rod’s centre.

(a) At what value of $\mathit{d}$are rod and particle stationary after the hit?

(b) In which direction do rod and particle rotate if $\mathit{d}$is greater than this value?

A top spins at $\mathbf{\text{30rev/s}}$about an axis that makes an angle of $\mathbf{\text{30°}}$with the vertical. The mass of the top is $\mathbf{\text{50kg}}$, its rotational inertia about its central axis is ${\mathbf{\text{5.0×10-4kg.m}}}^{\mathbf{\text{2}}}$, and its centre of mass is $\mathbf{4}\mathbf{.0}\mathit{c}\mathit{m}$from the pivot point. If the spin is clockwise from an overhead view,

(a) what are the precession rate?

(b) what are the direction of the precession as viewed from overhead?

Question: Figure gives the speed vversus time tfor 0.500 kg a object of radius 6.00 cmthat rolls smoothly down a 30 ⁰ramp. The scale on the velocity axis is set by v_s = 4.0 m/sWhat is the rotational inertia of the object?

The angularmomenta$\mathbf{l}\left(\mathbf{t}\right)$. of a particle in four situations are (1)$\text{l=3t+4}$(2)$\mathbf{}\mathbf{l}\mathbf{=}\mathbf{-}6{\mathbf{t}}^2$(3$\mathbf{l}\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}$(4)$\mathbf{l}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{/}\mathbf{t}\mathbf{}$In which situation is the net torque on the particle (a) zero, (b) positive and constant, (c) negative and increasing in magnitude (t>0), and (d) negative and decreasing in magnitude (t>0)?

Question: In Figure, a solid cylinder of radius 10 and mass 12 kgstarts from rest and rolls without slipping a distance L =6.0 mdown a roof that is inclined at the angle $\theta ={30}^0$. (a) What is the angular speed of the cylinder about its centre as it leaves the roof? (b) The roof’s edge is at height H = 5.0mHow far horizontally from the roof’s edge does the cylinder hit the level ground?

A rhinoceros beetle rides the rim of a horizontal disk rotating counterclockwise like a merry-go-round. If the beetle then walks along the rim in the direction of the rotation, will the magnitudes of the following quantities (each measured about the rotation axis) increase, decrease, or remain the same (the disk is still rotating in the counterclockwise direction): (a) the angular momentum of the beetle–disk system, (b) the angular momentum and angular velocity of the beetle, and (c) the angular momentum and angular velocity of the disk? (d) What are your answers if the beetle walks in the direction opposite the rotation?

Suppose that the yo-yo in Problem 17, instead of rolling from rest, is thrown so that its initial speed down the string is$\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$.(a) How long does the yo-yo take to reach the end of the string? As it reaches the end of the string, what are its (b) total kinetic energy, (c) linear speed, (d) translational kinetic energy, (e) angular speed, and (f) rotational kinetic energy?

Question: Figure shows the potential energy U (x) of a solid ball that can roll along an xaxis. The scale on the Uaxis is set by U_s =100j. The ball is uniform, rolls smoothly, and has a mass of 0.400 kgIt is released at x = 7.0 mheaded in the negative direction of the xaxis with a mechanical energy of 75 J (a) If the ball can reach x = 0 m, what is its speed there, and if it cannot, what is its turning point? Suppose, instead, it is headed in the positive direction of the xaxis when it is released at x = 7.0 m with 75J(b) If the ball can reach x = 13m, what is its speed there, and if it cannot, what is its turning point?

Figure 11-27 shows an overheadview of a rectangular slab that can

spin like a merry-go-round about its center at O. Also shown are seven

paths along which wads of bubble gum can be thrown (all with the

same speed and mass) to stick onto the stationary slab. (a) Rank the paths according to the angular speed that the slab (and gum) will have after the gum sticks, greatest first. (b) For which paths will the angular momentum of the slab(and gum) about Obe negative from the view of Fig. 11-27?