A bob of mass m = 0.300kg is suspended from a fixed point with a massless string

Question

A bob of mass m = 0.300kg is suspended from a fixed point with a massless string of length L = 25.0cm . You will investigate the motion in which the string traces a conical surface with half-angle ? = 20.0?.

What tangential speed v must the bob have so that it moves in a horizontal circle with the string making an angle 20.0? with the vertical?

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 20.0 degree from the vertical? Express your answer numerically in meters per second.

Explanation / Answer

Tension in the y direction.

T * cos 20 - mg = 0 N

T * cos 20 = mg

and the sum of the forces in the horizontal direction, there is the centripetal force and the component of tension in the x-direction.

T * sin 20 = m * (v^2 / R)

Rearrange the first equation to solve for T and plug it into the second one.

T = mg / cos 20

(mg / cos 20) * sin 20 = m * (v^2 / R)

Solve for V

V^2 = g * R * tan 20

v = sqrt (g * R * tan 20)

and we know that R = L * sin 20

V = sqrt (g * L * sin 20 * tan 20)

V = sqrt ( 9.8 m/s^2 * 0.25 m*sin20 . tan20 )

= 0.304 m/s

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