A bob of mass m is suspended from a fixed point with a massless string of length

Question

A bob of mass m is suspended from a fixed point with a massless string of length L (i.e., it is a pendulum). You are to investigate the motion in which the string moves in a cone with half-angle ?.

Conical Pendulum I (Figure 1) A bob of mass m is suspended from a fixed point with a massless string of length L (i.e, it is a pendulum). You are to investigate the motion in which the string moves in a cone with half-angle theta. Part A What tangential speed, v, must the bob have so that it moves in a horizontal circle with the string always making an angle theta from the vertical? Express your answer in terms of some or all of the variables m, L, and theta, as well as the acceleration due to gravity g. Part B How long does it take the bob to make one full revolution (one complete trip around the circle)? Express your answer in terms of some or all of the variables m, L, and theta, as well as the acceleration due to gravity g.

Explanation / Answer

Note that the sum of forces along y is          
          
Sum(Fy) = Ty - mg ---> Tcos(A) - mg = 0          
          
Thus, T = mg/cos(A)  
          
Thus, Tx = Tsin(A) = mgsin(A)/cos(A)

or

Tx = mgtan(A)

  
Here, Tx = ma_c, where a_c is the centripetal force.          
          
Thus,          
          
a_c = Tx/m = gtan(A)  
          
As a_c = v^2/R = gtan(A)

Then, v = sqrt [Rgtan(A)].           
          
Also, R = Lsin(A).
          
Thus,          
          
v = sqrt [Lgtan(A)sin(A)]   [ANSWER, PART A]

For part B, as T = 2 pi R / v, and R = Lsin(A), then

T = 2pi[Lsin(A)]/sqrt[Lgtan(A)sin(A)]

Simplifying,

T = 2 pi sqrt {[Lsin (A)]/[g tan (A)]}   [ANSWER]

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