centripetal forces

Question

A favourite swimming hole includes a rope swing from an elevated landing. The boy swings down and releases with a velocity of vf at the bottom of the swing. Use Energy, then Dynamics to determine the tension on the rope at the bottom, in terms of m and g.
Energy Method:



Dynamics Method:

Answer 1

The energy will give v_f as a function of initial conditions.  To get tension T, you need to use the dynamics method.

 

Suppose the rope has length L and is initially at an angle θ from the vertical.  Then by energy conservation,

 

mv_f²/2 = mgΔh = mg(L - Lcosθ) = mgL(1 - cosθ) ⇒

 

v_f² = 2gL(1 - cosθ)

 

At the bottom, the centripetal force is

 

mv_f²/L = T - mg

 

T = m(v_f²/L + g) = m[(1/L)2gL(1 - cosθ) + g]

 

T = mg(3 -2 cosθ)