Can someone explain this problem in quite a bit of detail, step bystep. I have a
ID: 1680875 • Letter: C
Question
Can someone explain this problem in quite a bit of detail, step bystep. I have asked this question once before and gotten two answersbut I'm not putting all the pieces together. Thankyou.
A uniform rod of mass m and length L is pivoted about an axlethrough one
end. The other end is attached to a horizontal massless springof
spring constant k. The spring is neither stretched nor compressedwhen
the rod hangs straight down. The bottom end of the rod is pulled tothe
right and released. The system oscillates in simple harmonicmotion. You
can assume the rod’s angle from the vertical is alwayssmall.
(a) Starting with Newton’s second law for rotation,write a differential equation for the position of therod as a function of time. i.e. write the equation of motionfor the system. (Hint:There are two restoring forces acting on the rod.)
(b) From the solution to your differential equation,determine the angular frequency of therod. (Hint: sincos=1sin(2))2
Explanation / Answer
a) 2 restoring torques: gravity mg, its torque = mg*(L/2)sin elastic force kx = k*(Lsin), its torque =kLsin*Lcos net torque = mg*(L/2)sin + kLsin*Lcos note this net torque is to make decrease, so the restoringtorque is - mg*(L/2)sin - kLsin*Lcos since is small, sin ˜ , cos˜ 1 net torque = -L(mg/2 + kL) =Id2/dt2, where I = mL2/3 d2/dt2 = -3(g/2L + k/m) b) solution: = Acos(t + ), where =[3(g/2L + k/m)] c) period = 2/ = 2/[3(g/2L + k/m)]