A simple pendulum consists of a light rod of length L = 0.9 m anchored at a fixed point (like the ceiling), and a small mass m = 0.5 kg attached to the free end.  The configuration is moved to an angle θ = 39 degrees from the vertical and released from rest.  Due to the presence of dissipative nonconservative forces (i.e. air resistance and friction at the anchor), 0.17 J of mechanical energy is removed from the system in its swing from θ to the vertical. i.e.  Wnc = -0.17 J. What is the tension (in Newtons) in the light rod at the instant the pendulum is vertical?

ΔE=0.17 J,

h=L(1-cosα)
PE=KE+ΔE
mgh=mv²/2 + ΔE
mv² =2{mgL(1-cosα)- ΔE}
T= mv²/L + mg = 2{mgL(1-cosα)- ΔE}/L +mg

Thank you! Very helpful.

To find the tension in the light rod at the instant the pendulum is vertical, we can use the principle of conservation of mechanical energy.

Let's start by defining the variables given in the problem statement:
- Length of the rod, L = 0.9 m
- Mass of the attached mass, m = 0.5 kg
- Angle from the vertical, θ = 39 degrees
- Work done by nonconservative forces, Wnc = -0.17 J

The initial mechanical energy of the system when the pendulum is released is the sum of the potential energy and kinetic energy:
E_initial = PE_initial + KE_initial
E_initial = -m * g * L * cos(θ) + (1/2) * m * L^2 * ω^2
Here, ω is the angular velocity, which is zero when the pendulum is released.

When the pendulum swings to the vertical position, all its energy is now in the form of potential energy:
E_final = PE_final
E_final = -m * g * L * cos(90 degrees)

According to the conservation of mechanical energy, the initial energy minus the work done by nonconservative forces should be equal to the final energy:
E_initial - Wnc = E_final

Substituting the expressions for E_initial and E_final:
-m * g * L * cos(θ) + (1/2) * m * L^2 * ω^2 - Wnc = -m * g * L * cos(90 degrees)

We know that cos(90 degrees) = 0, and ω = 0 since the pendulum is at rest when released. Therefore, the equation becomes:
-m * g * L * cos(θ) - Wnc = 0

Solving this equation for tension, we get:
Tension = (-m * g * L * cos(θ) - Wnc) / L

Plugging in the given values:
Tension = (-0.5 kg * 9.8 m/s^2 * 0.9 m * cos(39 degrees) - (-0.17 J)) / 0.9 m

After evaluating this equation, we get the tension in Newtons at the instant the pendulum is vertical.