91Ó°ÊÓ

A cannonball and a marble roll smoothly from rest down an incline. Is the cannonball’s (a) time to the bottom and (b) translational kinetic energy at the bottom more than, less than, or the same as the marble’s?

Question: In Figure, a solid brass ball of mass 0.280 g will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radiusr=14.o cm and the ball has radius r<<R(a) What is hif the ball is on the verge of leaving the track when it reaches the top of the loop? If the ball is released at height, h = 6.00R (b) What is the magnitude of the horizontal force component acting on the ball at point Q? (c) What is the direction of the horizontal force component acting on the ball at point Q?

A solid brass cylinder and a solid wood cylinder have the same radius and mass (the wood cylinder is longer). Released together from rest, they roll down an incline. (a) Which cylinder reaches the bottom first, or do they tie? (b) The wood cylinder is then shortened to match the length of the brass cylinder, and the brass cylinder is drilled out along its long (central) axis to match the mass of the wood cylinder.Which cylinder now wins the race, or do they tie?

Question: Non-uniform ball. In Figure, a ball of mass M and radius R rolls smoothly from rest down a ramp and onto a circular loop of radius 0.48 m The initial height of the ball is h =0.36m. At the loop bottom, the magnitude of the normal force on the ball is 2.00 mg. The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form $\mathit{I}\mathbf{=}\mathit{β}\mathit{M}{\mathit{R}}^{\mathbf{2}}$, but$\mathit{β}$is not 0.4as it is for a ball of uniform density. Determine $\mathit{β}$.

In Figure, a small, solid, uniform ball is to be shot from point P so that it rolls smoothly along a horizontal path, up along a ramp, and onto a plateau. Then it leaves the plateau horizontally to land on a game board, at a horizontal distance d from the right edge of the plateau. The vertical heights are${\mathit{h}}_{\mathbf{1}}\mathbf{=}\mathbf{5}\mathbf{.00}\mathbf{\text{ c³¾}}$and ${\mathit{h}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.60}\mathbf{\text{ c³¾}}$. With what speed must the ball be shot at point P for it to land at$\mathit{d}\mathbf{=}\mathbf{6}\mathbf{.00}\mathbf{\text{ c³¾}}$?

A bowler throws a bowling ball of radius $\mathit{R}\mathbf{}\mathbf{}\mathbf{=}\mathbf{11}\mathbf{}\mathbf{\text{cm}}$along a lane. The ball (in figure) slides on the lane with initial speed ${\mathit{v}}_{\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{,}\mathbf{0}}\mathbf{}\mathbf{}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{\text{m/s}}$and initial angular speed ${\mathit{Ó\neg }}_{\mathbf{0}}\mathbf{}\mathbf{=}\mathbf{0}$. The coefficient of kinetic friction between the ball and the lane is. The kinetic frictional force$\stackrel{→}{f_k}$acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed ${\mathit{v}}_{\mathbf{c}\mathbf{o}\mathbf{m}}$ has decreased enough and angular speed v has increased enough, the ball stops sliding and then rolls smoothly.

(a) What then is ${\mathit{v}}_{\mathbf{c}\mathbf{o}\mathbf{m}}$in terms ofV? During the sliding (b) What is the ball’s linear acceleration (c) What is the angular acceleration? (d) How long does the ball slide? (e) How far does the ball slide? (f) What is the linear speed of the ball when smooth rolling begins?

Non-uniform cylindrical object.InFigure, a cylindrical object of massMand radius Rrolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance d = 0.506m from the end of the ramp. The initial height of the object is H = 0.90m ; the end of the ramp is at height h = 0.10m . The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form $\mathit{l}\mathbf{=}\mathit{β}\mathit{M}{\mathit{R}}^{\mathbf{2}}$but b is not 0.5as it is for a cylinder of uniform density. Determine $\mathit{β}$.

A yo-yo has a rotational inertia of 950gcm² and a mass of 120g. Its axle radius is 3.2mm, and its string is 120cm long. The yo-yo rolls from rest down to the end of the string. (a) What is the magnitude of its linear acceleration? (b) How long does it take to reach the end of the string? As it reaches the end of the string, (c) What is its linear speed? (d) What is its translational kinetic energy? (e) What is its rotational kinetic energy? (f) What is its angular speed?

In 1980, over San Francisco Bay, a large yo-yo was released from a crane. The 116kg yo-yo consisted of two uniform disks of radius 32cmconnected by an axle of radius 3.2cm(a) What was the magnitude of the acceleration of the yo-yo during its fall ? (b) What was the magnitude of the acceleration of the yo-yo during its rise? (c) What was the tension in the cord on which it rolled? (d) Was that tension near the cord’s limit of 52kN? Suppose you build a scaled-up version of the yo-yo (same shape and materials but larger). (e) Will the magnitude of your yo-yo’s acceleration as it falls be greater than, less than, or the same as that of the San Francisco yo-yo? (f) How about the tension in the cord?

In unit-vector notation, what is the net torque about the origin on a flea located at coordinates $\left(0,-4.0\text{m},5.0\text{m}\right)$when forces $\stackrel{\mathbf{→}}{{\mathbf{F}}_{\mathbf{1}}}\mathbf{=}\mathbf{}\left(3.0\text{N}\right)\hat{\mathbf{k}}$and $\stackrel{\mathbf{→}}{{\mathbf{F}}_{\mathbf{2}}}\mathbf{=}\left(-2.0\text{N}\right)\stackrel{\mathbf{âˆ\S }}{\mathbf{J}}$act on the flea?