2.003J/1.053J Dynamics and Control I Fall 2007 Problem Set 3 Out: Monday, 24 September, 2007 Due: Monday, 1 October, 2007 1. Pulley problem A pulley of radius R is rigidly fixed to the wall as shown in the figure. A point mass is attached to one end of a non-elastic string of length lo and the other end of the string is fixed to the top of the pulley as shown in the figure. Initially the mass is held such that the string is tight and horizontal and is suddenly released. In the previous problem set we found the velocity and acceleration of the mass with respect to a stationary observer on the ground. Note that gravity acts. Find the equation of motion of the mass: a. Using Newton’s laws b. Using the work-energy principle lo g Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].2. Normal reaction due to Coriolis Effect A mass m lies in a straight radial channel dug on the surface of a circular, horizontal platform of radius R, as shown in the figure below. The platform rotates anticlockwise at a constant angular velocity ω about the vertical axis (coming out of the page through the center of the platform). The mass is attached to a string that is being pulled in from the center of the platform at a constant rate dr(t)/dt = -v. Find the normal force N = N(r) that the mass exerts on the wall of the channel. Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].3. Normal reaction due to Coriolis Effect A mass m lies in a straight channel dug on the surface of a circular, horizontal platform of radius R, as shown in the figure below. The center of the channel is at a distance d from the center of the platform. The platform rotates anticlockwise at a constant angular velocity ω about the vertical axis (coming out of the page through the center of the platform). The mass is free to slide in the channel and the contact surface is frictionless. Find the normal force that the mass exerts on the wall of the channel as a function of the distance of the mass from the center of the channel. d Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].4. Coefficient of restitution A ball of mass m is released from height Ho as shown in the figure below. It collides with the floor for the first time and bounces back and rises up to height H1 and again drops down. The second time it collides with the floor and again bounces back to height H2 and so on… The coefficient of restitution in each collision is c, where c is strictly less than 1. Thus, after each collision, the height up to which the ball rises back is reduced. • What is the relation between Ho and H1? • As time tends to infinity, how many times the ball would have bounced off the floor? • What is the total distance traveled by the ball? Ho g H1 H2 Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
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