Start from basic physics principles , and include explanations such that another

Question

Start from basic physics principles, and include explanations such that another student would be able to follow and understand your reasoning. ANSWER ALL PARTS TO RECIEVE BEST ANSWER VOTE!

Fog Saute 2 A mass m oonstrained to move on a frictionless horizontal sutace is attached to a frictionless peg by an ideal, massiless spring with spring constant k. The unstretched length of the spring is , as shown in Figure 1. When the mass moves in a circle about the peg withconaut angular velocaya,the length of the pring 4asshown Figure 2 Express you answers to partsa -d in tems of some or all of the quantities mand i, and any necessary (a) Determine the length ) Assume heocal energy of the system in Figuue 1 is zeoDetermine the total eney of the rotating system in Figure 2 (c) Determine the magnitade of the angular momentum of the ystemm, small while the mass is rounag about the peg with mular velocity a, it is struck by a haamer that provides impulse directed inward towaad the peg d) Determine the torque exerted on the mass by this hammer stnke. (e) On the axes below, sketch graphs to indicate qualitaoively the manner in which the length of the spring and the angular velocity a will vay with time in the subsequent mocioe You may either skeach she grapls on the aves provided below, or reproduce the axes wilah your work

Explanation / Answer

When moving in circle,

the centripetal force necessary to move in circle must be provided by the spring force

Now, spring force, Fs = k*x

where x = extension of spring = L2 - L1

So, Fs = k*(L2 - L1)

Now, centripetal force, Fc = m*W^2*R

where R = radius of circle = L2

So, Fc = m*W^2*L2

For circular motion, Fc = Fs

So, m*W^2*L2 =  k*(L2 - L1) ----------- (1)

So, m*W^2/k = (L2-L1)/L2 = 1 - L1/L2

So, L1/L2 = 1 - mW^2/k

So, L2 = L1/(1 - mW^2/k) = k*L1/(k - mW^2) <----------answer

b)

Total energy = Spring Potential energy + Kinetic energy of mass = Up + KE

Up = 0.5*k*x^2 = 0.5*k*(L2-L1)^2

KE = 0.5*m*W^2*R^2 = 0.5*m*W^2*(L2)^2

So, TE =  0.5*k*(L2-L1)^2 + 0.5*m*W^2*(L2)^2

So, TE = 0.5*k*L1*m*W^2*(mW^2+k)/(k-mW^2)^2 <--------answer

c)

angular momentum = m*W*L2^2 = m*W*(k*L1/(k - mW^2))^2 <----------answer

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