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2.032 DYNAMICS Fall 2004 Problem Set No. 1
Out: Wednesday, September 15, 2004 Due: Wednesday, September 22, 2004 at the beginning of class
Problem 1 (Doctoral Exam, 1999) A pendulum is constructed by attaching a mass m to an extensionless string of fixed length l. The upper end of the string is connected to the uppermost point of a vertical fixed disk of radius R (R < l/π), as shown below. At t = 0 the mass hangs at rest at the equilibrium position θ = 0, when it is given an initial velocity v 0 along the horizontal. Derive expressions for the two extreme deflections (in terms of θ) of the pendulum resulting from this initial perturbation. Do not make a small-angle approximation.
g
v 0
R
m
θ
Courtesy of Prof. T. Akylas. Used with permission.
Problem 2
A point mass moves without friction on a horizontal plane. A massless inextensible string is attached to the point mass and led through a hole (see figure below). At time = t 0 the mass moves along a circle with constant velocity v 0. We gradually pull the free end of the string downwards until, at time t 1 , we have |r(t 1 )| = L 0 /2. What is the velocity of the mass at time t 1?
Plane (no friction)
r(t)
String
m
O
Problem 4 (adapted from Crandall et al., 2-35) Consider the system shown below under the assumption that the pendulum arm connect- ing m 2 and m 3 is massless. By applying momentum principles, obtain the differential equations of motion for the generalized coordinates x 1 , x 2 and θ.
m 3
θ
m 2
x 1 x 2 k 1 k 2 m 1
g
Problem 5 (Doctoral Exam, 1999) Two identical rods of length l, that have equal masses m attached at their ends, are clamped at an angle θ to a shaft as shown. (The shaft and the rods are in the same plane.) What reaction forces must the bearings be able to withstand, if the angle θ can be set anywhere from zero to 90◦^ and the maximum angular velocity of the shaft is ω? (For simplicity, you may neglect the mass of the rods and ignore the effects of gravity.)
m
l
d
l
m
Courtesy of Prof. T. Akylas. Used with permission.
