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8.01x Classical Mechanics, Fall 2016 Massachusetts Institute of Technology
Problem Set 11
- Bug Walking on Pivoted Ring A ring of radius R and mass m 1 lies on its side on a frictionless table. It is pivoted to the table at its rim. A bug of mass m 2 walks on the ring with constant speed v relative to the ring, starting at the pivot, when the ring is initially at rest. Take ˆk to point out of the page.
(a) What is the angular velocity of the ring when the bug is halfway around? Express you answer in terms of some or all of the following: m 1 , m 2 , v, R and ˆk. (b) What is the angular velocity of the ring when the bug is back at the pivot? Express you answer in terms of some or all of the following: m 1 , m 2 , v, R and kˆ.
- A Rigid Rod
A rigid uniform rod of length d and mass m is lying at rest on a horizontal frictionless table and pivoted at the point P. A point-like object of mass m is moving to the right with speed v. It collides and sticks to the rod at a distance 2d/3 from the pivot. A second point-like object of mass m is moving to the left (see figure) with speed v and collides with the rod at exactly the same instant as the first particle at a distance d/ 3 from the pivot. The moment of inertia of a rod for axis through the center of mass and perpendicular to the plane of the rod is Icm = 121 md^2. After the collision, the rod and the two particles all rotate about the pivot point with angular speed ωf.
(a) What is the component of the angular speed ωf of the two particles and the rod immediately after the collision? Express your answer in terms of d, m, and v, as needed. Assume clockwise (into the page) to be positive. (b) What is the ratio of the change in kinetic energy to the initial kinetic energy of , Kf^ −Ki^? Expre the system (^) Ki ss your answer in terms of d, m, and v, as needed.
- Elastic Collision of Object and Pivoted Ring
A rigid hoop of radius R and mass mR is lying on a horizontal frictionless table and pivoted at the point P (shown in the figure below). A point-like object of mass m is moving to the right with speed v 0. It collides elastically with the hoop at its midpoint. After the collision, the object is moving with an unknown speed vf to the left and the hoop rotates counterclockwise about its pivot point with angular speed ωf. The moment of inertia of a hoop for axis through the center of mass and perpendicular to the plane of the hoop is Icm = mRR^2.
What is the speed vf of the object immediately after the collision? Express your answer in terms of R, m, mR, and v 0 as needed (do not use ωf in your answer).
- A Spaceship and a Planet
Spaceship 1 has mass m 1 and is moving with speed v 1 in a circular orbit of radius R around a planet of mass mp. Spaceship 2 has mass m 2 and is moving in an elliptical orbit around the same planet. The mass of the planet is much, much greater than the mass of either spaceship. When spaceship 2 is at its furthest distance 3R from the planet, it is moving with speed v 2. When spaceship 2 is at its closest distance R from the planet, it is moving with speed vp. The two spaceships are orbiting in the same plane as shown in the figures above. At a later time, both spaceships arrive nearly simultaneously at a point corresponding to the closest approach of spaceship 2. Spaceship 2 fires its rockets in order to reach the same speed v 1 as spaceship 1 in order to dock together. You may assume that the elapsed time interval for docking is very small compared to the orbital periods of the spaceships. Let G be Newton’s universal constant of gravity. What is the change in the speed, ∆v = v 1 − vp, of spaceship 2 in order for the two spaceships to dock together? (Does spaceship 2 speed up or slow down in order to dock?) Express your answer only in terms of G, R and mp.
