Rotational Kinetic Energy
A ball and a hoop of equal mass and radius start side by side and proceed to roll down an incline. Which reaches the bottom first? Explain.
The ball; start with same kinetic energy, mass is closer to the central axis, lower rotational inertia, greater angular acceleration
A rotating wheel is supported by a fixed rod perpendicular to the plane of the wheel and through the geometric center/center of mass of the wheel. What is the direction of the wheel's angular momentum if there is a force on the wheel at 9 o-clock pointing up at about 45 degrees. (Hint: First determine direction of torque)
Into the page. Angular momentum is in the same direction as the angular acceleration vector (due to L=Iw), which is in the same direction as torque (as t=Ia). Using rxF, torque is into the board and therefore so is angular momentum.
Can an object that does not move in a circular path still be defined with angular momentum? If yes, which formula would be "most directly" used here?
Yes, angular momentum of a point not moving in a circular path can still be defined relative to any reference point. L=rmv⟂
Consider a disk rolling down an incline without slipping. The point of contact of the object with the surface is _______ at rest. The contact point _____ be moving relative to the surface.
instantaneously, must not
For an object to be in static equilibrium, what two things must be zero?
Net force and net torque.
If you give a roll of relatively firm, "disconnected," toilet paper an initial push on a flat, horizontal, hardwood floor, it may not slow down and come to a rest as expected but rather pick up speed. How?
As the roll lays down more toilet paper rolls out and so its center of mass lowers. As the c.o.m drops, the gravitational potential energy is converted into kinetic energy and the roll may get faster.
A rotating ice skater has 100 joules of rotational kinetic energy. The skater increases her moment of inertia by a factor of 2, how will her rotational speed change?
Angular Momentum must be conserved, so if I increases by a factor of 2, w must decrease by a factor of 2.
A spinning ice skater with their arms stretched outward has kinetic energy, angular velocity, and angular momentum. If the skater pulls their arms in, which of those quantities will be conserved and how are the non-conserved ones effected (go up, go down, etc.)
Angular Momentum is conserved, no torque about axis of rotation; Increase in angular velocity, moment of inertia is decreasing and since L is constant and I is going down w has to increase to counteract; Increase in rotational kinetic energy, formula.
Which physical law states that the square of a plane'ts orbital period is proportional to the cube of the semi-major axis of the planets orbit?
Kepler's Third Law of Planetary Motion
If Mr. Burmester and his bike were rolling down a *steep* hill, would it be preferable for him to sit straight or sit leaning forward. Please explain why.
Leaning forward bring the center of mass closer to the ground and causes more stability - as well as minimizing torque - BUT sitting straight would be better for the plot 🫡, make life interesting.
A cylinder of mass M and radius R rolls (without slipping) down an inclined plane (of height h and length L) whose incline angle with the horizontal is a. Determine the linear speed of the cylinder's com when it reaches the bottom of the incline, assuming it started from rest at the top, 10.
v=sqrt((4/3)gh). First set initial energy equal to final energy and simplify.
A string threaded through a whole in a frictionless table is attached to a puck. the puck is set in motion so that it circles around the hole. The strong is pulled, decreasing the puck's radius of motion. When this happens, the puck's angular velocity increases, explain this using angular momentum.
Since there are no external torques acting on the puck, angular momentum is conserved. The moment of inertia is I=mr^2 (think of it as a point mass), so as r decreases so does I. To keep L constant, w must therefore increase.
A child of mass m=30kg stands at the edge of a small merry-go-round that's rotating at a rate of 1 rad/s. The merry-go-round is a disk of radius R=2.5m and mass M=100kg. If the child walks in towards the center of the disk and stops 0.5m from the center, what will happen to the angular velocity of the merry-go-round (assuming friction can be ignored, 11)?
w=1.6rad/s. Angular momentum is conserved, the child does not provide external torque. Find the initial and final rotational inertia's, ((1/2)M+m)R^2 and (1/2)MR^2+mr^2 respectively. Set Li=Lf and solve.
A cylinder of mass M and radius R rolls (without slipping) down an inclined plane whose incline angle with the horizontal is a. Determine the minimum coefficient of friction that will allow the cylinder to roll without slipping on the incline. (Hint: determine the acceleration of the com first, 9).
Minimum coefficient of STATIC friction =(1/3)tan(a). First, draw a free body diagram and use the parallel axis theorem to determine a=(2/3)gsin(a). Then sum the forces with Newton's Second Law to solve.