question archive A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod
A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod
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A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod. A massless string (length i) attached to the ball runs over a massless. frictionless pulley and supports a block of mass M. as shown in Figure 4.27. The positions of the two masses can be speci?ed by the one angle 6. (3) Write down the potential energy U (9). {The PE is given easily in terms of the heights shown as it and H. Eliminate these two variables in favor of 19 and the constants b and I. Assume that the pulley and ball have negligible size.) (b) By differentiating U (19] ?nd whether the system has an equilibrium position. and for what values of m and M equilibrium can occur. Discuss the stability of any equilibrium positions.
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Step-by-step explanation
") U(D ) = agh + Man - 0 8 we can write , H = 1-b Sino also, tang : b = ) h = b (3 tano From 1, (2 ) 2 3 Ulolz mab + Mg / 1- b land Sine = mgblot + ing ( 1- bcoseco ) b ) Jule ) = mab ( - losec '0) + my (-bx (- loto (seco) ) do = - mablosec + mab losecowto. For equilibrium duce) 20 Mabloseco coto = myblseco M LOVO = m
3 3 So for 0 = lou"l mthou is equilibrium as we
know , So For m <M equilibrium will occur . gt is a
stable equilibrium Position
Please see attached file
