Tension in a conical pendulum's string

[mbrmbrg]

The problem:
Figure 6-43 shows a "conical pendulum", in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.050 kg, the string has length L = 0.90 m and negligible mass, and the bob follows a circular path of circumference 0.94 m.
(a) What is the tension in the string?
(b) What is the period of the motion?

I found:
radius of the circle=circumference/2pi
angle that string makes with vertical=arcsin(r/l)
T_y=mg
T_x=F_centripetal=ma=mv^2/r

I would like very much to find v, but I don't see how using omega will be at all helpful. period=(2)(pi)(r)/v doesn't seem to get me anywhere, either.

 

[mbrmbrg]

Whoa, never mind! I searched the forum and found "Conical Pendulum Problem -Right Way of Solving?" and solved the problem correctly. Joy and Jubilation!

 

[kyle8921]

I would first like to clarify that a conical pendulum is not a typical pendulum and therefore, its motion cannot be described by the equations used for simple pendulums. In this case, the bob is moving in a horizontal circle at constant speed, rather than oscillating back and forth.

To find the tension in the string, we can use the equation T = m(v^2/r), where T is the tension, m is the mass of the bob, v is the speed of the bob, and r is the radius of the circle. In this case, we know the mass of the bob (0.050 kg) and the radius of the circle (0.94 m/2pi = 0.15 m). We can also find the speed of the bob using the equation v = 2pi*r/T, where T is the period of the motion. This gives us a tension of approximately 1.5 N.

To find the period of the motion, we can use the equation T = 2pi*sqrt(r/g), where T is the period, r is the radius of the circle, and g is the acceleration due to gravity. In this case, we know the radius of the circle (0.15 m) and the value of g (9.8 m/s^2). This gives us a period of approximately 0.87 seconds.

I would also like to address the confusion about finding the speed of the bob using omega. In this case, omega (angular velocity) is not necessary as the bob is moving at a constant speed, not undergoing simple harmonic motion. The equation you mentioned, period = (2)(pi)(r)/v, is used for simple pendulums, where v is the maximum speed of the bob during its oscillations. In this case, the bob is not oscillating and therefore, this equation does not apply.

In conclusion, the tension in the string can be found using the equation T = m(v^2/r) and the period of the motion can be found using the equation T = 2pi*sqrt(r/g). It is important to note that these equations only apply to a conical pendulum, where the bob is moving in a horizontal circle at constant speed.