Question

A slender, uniform metal rod of mass M and length l is pivoted without friction about an axis through its midpoint and perpendicular to the rod. A horizontal spring, assumed massless and with force constant k, is attached to the lower end of the rod, with the other end of the spring attached to a rigid support. (Figure 1)

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2. Find the torque τ due to the spring. Assume that θ is small enough that the spring remains effectively horizontal and you can approximate sin(θ)≈θ (and cos(θ)≈1).

Express the torque as a function of θ and other parameters of the problem.

τ= _____________

3. What is the angular frequency ω of oscillations of the rod?

Express the angular frequency in terms of parameters given in the introduction.

ω= _______

Answer 1

Note: There is no torque due to gravity on rod. @ - T = ex T = rF sind in F = -xx K 9 니 그 From geometry ordt 90 = 180 1. $ = 90° - - © . e4 соилет - rF sin (90-0) T = r F cos o - 3 From Hook's law F = kx. - Also r = 2 and for small o caso=1 .: T = - Lux - 4 Also sin e = a (From figure) sin D = 22 for small angles sin o=0;. 02 21 42

and a zlo 2 Putting 1 this value in 3 T = -Lk/ T = - KL²o Answer (6) T = IX I As dad²o M 12 at2 For euod I = M12 . T = M1² a As d = a 20 in T = M2² ( 120 ) -KL20 = ML2 2e ['Tz - KLO 1. - 3K o ad20 at? | 1 | 2 cº. M Comparing it with W z 3K dt² who= do we get V M Auswer

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