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In physics class, I learned that the reason a pendulum moves about the equilibrium position is due to the force acting towards the center, which is the resultant force of tension and weight.

However, since tension force is orthogonal to the direction, shouldn't only gravity act as the object's restoring force about the equilibrium position? What is the role of the string's tension in simple harmonic motion, and does the horizontal component of tension have anything to do with the restoring force mgsinθ?

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It depends on what you're keeping track of. Typically we keep track of the angular displacement of the pendulum. In this case it's better to think about the torque acting on the pendulum about the point of rotation. Therefore, the tension force doesn't play a role in restoring the angular displacement to $0$ because it has no torque about the point of rotation. In other words, we have $$\tau=-mgr\sin\theta$$

However, if you are just interested in the horizontal position of the pendulum, then the tension force does matter. Splitting the forces up into horizontal and vertical components we have $$F_x=-T\sin\theta$$ $$F_y=T\cos\theta-mg$$ The tension force is the only force that has a horizontal component. Of course, this is more complicated than the above case, so you don't usually see this system analyzed in this way.

A mix between these two cases is to consider forces along the path and perpendicular to the path. Then we have $$F_r=mg\cos\theta-T$$ $$F_\theta=-mg\sin\theta$$ And I think this gets at the heart of what the tension force's role is. It's serves to constrain the motion of the pendulum to maintain its circular path. i.e. since the radius of the circle is not changing, it must be that $F_r=-mr\omega^2$, so the tension keeps the motion circular by bring equal to $mg\cos\theta+mr\omega^2$. Also, note that we get back to the first case by using the fact that $\tau=F_\theta r$.

In any case, as you can see there are multiple ways to look at the system. None of the ways are wrong, but some are more useful than others. It's hard to really say what any of the forces' role is in terms of restoring to equilibrium. Sure, gravity is the only force that has a component along the direction of motion, but the tension force is just as important to the overall motion of the pendulum.

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enter image description here

Look at this pendulum diagram. the pendulum can rotate about the point A. taking the sum of the torques about point A, you get:

$$\sum_\tau=I_p\,\ddot{\theta}+m\,g\,h=0\quad,(h=L\,\sin(\theta))$$

where dot mean time derivative and $I_p=m\,L^2$ is the pendulum inertia.

so the tension force T doesn't affect the equation of motion.

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