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This course includes collaboration policy, collision, conservation law, drag force, mass calculation, multiple stage rocket, estimates and uncertainties, Newton laws, potential energy, torque, friction, gravitational force, masses and rod, orbital velocity. This exam includes: Frictionless, Surface, Sliding, Crate, Coefficient, Critical, Force, Constant, Horizontal, Displacement, Moment, Inertia
Typology: Exams
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Sliding a crate. (15 points)
A mover is trying to slide a uniformly filled crate of length L and height H across the floor. There is friction with coefficient μ between the crate and the floor. The mover exerts a horizontal force F at the upper back edge of the crate. If μ > μ 0 , the crate will tip over before it slides. Calculate the critical friction coefficient μ 0. Gravity is directed downward.
Reel of tape on a frictionless surface. (15 points)
A long length of tape is wrapped around a reel which is at rest on a frictionless surface. The tape itself has negligible mass, but the reel has mass M, radius R, and moment of inertia I 0 about its center. The end of the tape is pulled horizontally with a constant force F , as shown in the figure. Calculate x, the horizontal displacement of the reel from its initial position when a length s of tape has unwound from the reel.
Compound gyroscope. (15 points)
A compound gyroscope consists of four identical flywheels mounted at the midpoints of the edges of a rigid square frame of length L. Each flywheel has moment of inertia I 0 and is spinning rapidly at the same angular velocity ω, with the sense of rotation shown in the figure. The entire assembly (frame plus flywheels) has total mass M, and the center of mass is at the geometric center of the square frame. One corner of the frame is attached to a frictionless pivot, so that the entire frame is free to rotate about the pivot in any direction. In the top-view figure, gravity is directed into the page.
(a) (10 points) Find the frequency of uniform precession Ω and indicate its direction.
(b) (5 points) Now suppose the entire compound gyroscope is placed inside an elevator. The elevator accelerates upward (opposite to gravity, so out of the page in the figure) with uniform acceleration A. Find the new frequency of uniform precession.
Particle in a central force field. (15 points)
A mass m moves under the influence of an attractive central force Ar^4 with angular momentum L, where A and L are positive constants. Define the potential energy to be zero at the origin. For what total energy will the motion be circular, and what is the radius of this circular orbit?
Projectile near two gravitating spheres. (15 points)
Two identical spheres of mass M and radius R are held fixed at a separation distance D, as shown. A small projectile of mass m is fired with initial velocity v 0 from the surface of one sphere directly toward the other. What is the minimum value of v 0 such that the projectile just reaches the second sphere? You should assume that the projectile is attracted gravitationally by each sphere, and that no other forces are acting.
