Question Part Points Submissions Used A conical pendulum consists of a bob (mass

Question

Question Part Points Submissions Used A conical pendulum consists of a bob (mass m) attached to a string (length L) swinging in a horizontal circle. As the string moves, it sweeps out the area of a cone. The angle that the string makes with the vertical is .

(a) What is the tension in the string. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration due to gravity.)

(b) What is the period of the pendulum. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration due to gravity.)

Explanation / Answer

a)
As the pendulum swings in a horizontal circle, there is no total acceleration vertically.

Let T be the tension in the string.

gravitational force vertically = mg
tension force vertically = T cos(phi)

there are no other forces, therefore T cos(phi) = mg

Therefore T = mg / cos(phi)

b)
The pendulum is swinging in a circle, therefore can use centripetal force equation:

F = m L sin(phi) v²

where v is the angular velocity of the pendulum.

The force F is provided by the tension in the string.

tension force horizontally = T sin(phi) = mg tan(phi)

mg tan(phi) = m L v² sin(phi)

v = (g / (L cos(phi)))

v = 2 / t, where t is the period of the pendulum

2 / t = (g / (L cos(phi)))

t = 2 (L cos(phi) / g)

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