Auniform rod of mass m and length L is pivoted about an axle throughone end. The
ID: 1680450 • Letter: A
Question
Auniform rod of mass m and length L is pivoted about an axle throughone end. The other end is attached to a horizontal massless springof spring constant k. The spring is neither stretched norcompressed when the rod hangs straight down. The bottom end of therod is pulled to the right and released. The system oscillates insimple harmonic motion. You can assume the rod’s angle fromthe vertical is always small.
(a)Starting with Newton’s second law for rotation, write adifferential equation for the position of the rod as afunction of time. i.e. write the equation of motion for the system.(Hint: There are two restoring forces acting on therod.)
(b) From the solution to your differential equation,determine the angular frequency
oftherod. (Hint: sincos=1sin(2))2
(c)Determine the period of oscillation of the rod.
Explanation / Answer
(a) Starting withNewton’s second law for rotation, write a differentialequation for the position of the rod as a function of time.i.e. write the equation of motion for the system. (Hint: Thereare two restoring forces acting on the rod.)
the two restoringforces on the rod are the spring force and gravitationalforce
the torque due tothese two forces isk*l*lcos+mg*l/2sin=I
where I is the moment ofinertia about the axle and is the angular acceleration=d2/dt2
this completes thedifferential equation
(b) From thesolution to your differential equation, determine the angularfrequency
oftherod.(Hint: sincos=1sin(2)) 2
we have =k*l*lcos+mg*l/2
(c) Determine the periodof oscillation of the rod.
pe4riod of oscillationof the rod is 2/
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