To find the minimum angle θ at which the pendulum must be released for the ball to go over the top of the peg without the string going slack, we can use the conservation of mechanical energy.
When the pendulum is released from an initial height h above the lowest point, its total mechanical energy at that point consists of its gravitational potential energy and its kinetic energy:
Gravitational Potential Energy (U): U = mgh
Kinetic Energy (K): K = 0 (at the highest point, the velocity is zero)
At the highest point, all of the pendulum's initial gravitational potential energy is converted into kinetic energy.
For the pendulum to clear the peg without the string going slack, the height of the highest point should be at least equal to the height of the peg (h = L/3).
So, when the pendulum reaches its highest point, the gravitational potential energy is equal to the height of the peg:
mgh = mg(L/3)
The mass (m), gravitational acceleration (g), and h can be canceled out:
gh = L/3
Now, we can solve for the height (h) using trigonometry. At the highest point, the vertical displacement of the ball is equal to L - Lcosθ, where θ is the angle of release. Setting this equal to L/3:
L - Lcosθ = L/3
Now, solve for cosθ:
L - L/3 = Lcosθ
(2/3)L = Lcosθ
cosθ = 2/3
Now, find θ:
θ = cos^(-1)(2/3)
θ ≈ 48.19 degrees
So, the pendulum must be released at an angle of at least approximately 48.19 degrees for the ball to go over the top of the peg without the string going slack.
