#Physical intuitiion I imagine that $\theta=\pi$ is a stable equilibrium and $\theta=0$ will be an unstable equilibrium. So the torque should be zero at $\theta=0$ and slowly (linearly) increase for small $\theta$, but in a direction which drives the system away from equilibrium. So $\tau \sim k \, \sin \theta$ seems like a reasonable conjecture for perturbations from the equilibria.

Find the potential energy of the system

Consider a small element in rod1 and find it's potential energy due to rod 2. Thsi will involve an integration over rod2. Now, integrate over rod1 to find the total potential energy of the system. Both integrations are over scalars... no mess with multiple components and projections. You will get an expression for the system's potential energy as a function of the angle between the rods.

#Finding the torque Fix your coordinate sustem with the origin at the hinge and (say) X-axis along rod2. Then, effectively, you only have 1 degree of freedom: the angle $\theta$ between the rods. To find the torque $\tau$, differentiate the expression for potential energy w.r.t $\theta$.

If you want to find the force, you could invert the expression $\int_0^L F(r) \, r\, dr = \tau$ to find $F(r)$ and then integrate that along the rod to find the tangential component of the force.

Physical intuitiion

I imagine that $\theta=\pi$ is a stable equilibrium and $\theta=0$ will be an unstable equilibrium. So the torque should be zero at $\theta=0$ and slowly (linearly) increase for small $\theta$, but in a direction which drives the system away from equilibrium. So $\tau \sim k \, \sin \theta$ seems like a reasonable conjecture for perturbations from the equilibria.

Find the potential energy of the system

Consider a small element in rod1 and find it's potential energy due to rod 2. Thsi will involve an integration over rod2. Now, integrate over rod1 to find the total potential energy of the system. Both integrations are over scalars... no mess with multiple components and projections. You will get an expression for the system's potential energy as a function of the angle between the rods.

Finding the torque

Fix your coordinate sustem with the origin at the hinge and (say) X-axis along rod2. Then, effectively, you only have 1 degree of freedom: the angle $\theta$ between the rods. To find the torque $\tau$, differentiate the expression for potential energy w.r.t $\theta$.

If you want to find the force, you could invert the expression $\int_0^L F(r) \, r\, dr = \tau$ to find $F(r)$ and then integrate that along the rod to find the tangential component of the force.

#Physical intuitiion I imagine that $\theta=\pi$ is a stable equilibrium and $\theta=0$ will be an unstable equilibrium. So the torque should be zero at $\theta=0$ and slowly (linearly) increase for small $\theta$, but in a direction which drives the system away from equilibrium. So $\tau \sim k \; sin \theta$ seems like a reasonable conjecture for perturbations from the equilibria.

Find the potential energy of the system

Consider a small element in rod1 and find it's potential energy due to rod 2. Thsi will involve an integration over rod2. Now, integrate over rod1 to find the total potential energy of the system. Both integrations are over scalars... no mess with multiple components and projections. You will get an expression for the system's potential energy as a function of the angle between the rods.

#Finding the torque Fix your coordinate sustem with the origin at the hinge and (say) X-axis along rod2. Then, effectively, you only have 1 degree of freedom: the angle $\theta$ between the rods. To find the torque $\tau$, differentiate the expression for potential energy w.r.t $\theta$.

If you want to find the force, you could invert the expression $\int_0^L F(r) \, r\, dr = \tau$ to find $F(r)$ and then integrate that along the rod to find the tangential component of the force.

#Physical intuitiion I imagine that $\theta=\pi$ is a stable equilibrium and $\theta=0$ will be an unstable equilibrium. So the torque should be zero at $\theta=0$ and slowly (linearly) increase for small $\theta$, but in a direction which drives the system away from equilibrium. So $\tau \sim k \, \sin \theta$ seems like a reasonable conjecture for perturbations from the equilibria.

Find the potential energy of the system

Consider a small element in rod1 and find it's potential energy due to rod 2. Thsi will involve an integration over rod2. Now, integrate over rod1 to find the total potential energy of the system. Both integrations are over scalars... no mess with multiple components and projections. You will get an expression for the system's potential energy as a function of the angle between the rods.

#Finding the torque Fix your coordinate sustem with the origin at the hinge and (say) X-axis along rod2. Then, effectively, you only have 1 degree of freedom: the angle $\theta$ between the rods. To find the torque $\tau$, differentiate the expression for potential energy w.r.t $\theta$.

If you want to find the force, you could invert the expression $\int_0^L F(r) \, r\, dr = \tau$ to find $F(r)$ and then integrate that along the rod to find the tangential component of the force.