For a conical pendulum (string length L, angle θ with vertical, mass m, velocity v), which set of force component equations is correct?

For a conical pendulum (string length L, angle θ with vertical, mass m, velocity v), which set of force component equations is correct?

• A. Vertical: T cos θ = m g; Horizontal: T sin θ = m v^2 / r, with r = L sin θ
• B. Horizontal: T cos θ = m g; Vertical: T sin θ = m v^2 / r, with r = L sin θ
• C. Vertical: T sin θ = m g; Horizontal: T cos θ = m v^2 / r, with r = L sin θ
• D. Vertical: T cos θ = m g; Horizontal: T sin θ = m v^2 / r, with r = L cos θ

Answer: (C)

Key idea: the tension along the string provides two components, and the vertical component must balance gravity while the horizontal component provides the centripetal force for the circular motion. Since the string makes angle θ with the vertical, the vertical component of tension is T cos θ and the horizontal component is T sin θ. The mass is in vertical equilibrium, so the vertical forces balance: T cos θ = m g. Horizontally, the horizontal component of tension pulls toward the axis and must supply the centripetal force for the circular path. That gives: T sin θ = m v^2 / r, where the radius of the circle is r = L sin θ (the horizontal projection of the string). So the correct set is vertical: T cos θ = m g; horizontal: T sin θ = m v^2 / r, with r = L sin θ. The other options mix up the sine and cosine components, which doesn’t match the geometry for a conical pendulum.

Key idea: the tension along the string provides two components, and the vertical component must balance gravity while the horizontal component provides the centripetal force for the circular motion.

Since the string makes angle θ with the vertical, the vertical component of tension is T cos θ and the horizontal component is T sin θ. The mass is in vertical equilibrium, so the vertical forces balance: T cos θ = m g.

Horizontally, the horizontal component of tension pulls toward the axis and must supply the centripetal force for the circular path. That gives: T sin θ = m v^2 / r, where the radius of the circle is r = L sin θ (the horizontal projection of the string).

So the correct set is vertical: T cos θ = m g; horizontal: T sin θ = m v^2 / r, with r = L sin θ. The other options mix up the sine and cosine components, which doesn’t match the geometry for a conical pendulum.