Mastering Physics Solutions: A Conical Pendulum

A Conical Pendulum

Part A = 3.21 N
Part B = 31.24 rpm

Solution Below:

A conical pendulum is formed by attaching a 0.300 kg ball to a 1.00 m-long string, then allowing the mass to move in a horizontal circle of radius 40.0 cm. The figure (Intro 1 figure) shows that the string traces out the surface of a cone, hence the name.

Part A

What is the tension in the string?

Start by finding the angle of the string, θ.

sin(θ) = 0.40 / 1.0
sin(θ) = 0.4
θ = 23.578°

Now find tension:

Tcos(θ) = mg
T * cos(23.578) = 0.3 * 9.8
T * 0.9165 = 2.94
T = 3.21 N

3.21 N

Part B

What is the ball’s angular velocity, in rpm?

You can find the angular velocity using the formula:

Tsin(θ) = mrv^2

θ (from Part A) is 23.57°, and you already found tension to be 3.21 N, so you can solve:

Tsin(θ) = mrv^2
3.21 * sin(23.57) = 0.3 * 0.4 * v^2
1.2836 = 0.12 * v^2
v^2 = 10.70

v = 3.271

Since the above is in radians per second, convert to rpm:

rpm = 3.271 * 60 / (2π)

rpm = 31.24

31.24 rpm