Calculate net force on the pendulum

In this image you can see a swinging pendulum and that there's a net force $$A$$ that causes it to move in circular motion. It is constantly updating velocity vector $$v$$ direction to be perpendicular with the direction towards the pivot point.

As I understand the net force $$A$$ should be $$A = T - mg$$, where $$T$$ is the tension force and is equal to $$T = mg\cos\theta$$. I tried to solve for net force $$A$$ but I'm not getting the correct result. In my calculations when $$\theta = 0$$, the net force is zero. But that's wrong because there's clearly always a force that affects the pendulum, since its velocity vector constantly changes its direction.

What am I missing?

You are missing concepts.

In a pendulum, the net force at any instant can be resolved into radial and tangential components(They are infact mutually perpendicular to each other at each instant). The radial component is along the string, directed towards the centre of circular motion. It actually provides the required centripetal acceleration . When the string makes angle $$\theta$$ from the vertical , you can write: $$T-mg cos\theta=\frac {mv^2}{l}$$

Note that at the instant of maximum displacement only, the pendulum instantaneously comes to rest. So at maximum angular displacement $$\theta_0$$, we can write:

$$T_0=mgcos\theta_0,$$ where $$T_0$$ is the tension in the string at that instant.

Note that the tension also varies accordingly throughout the motion.

The tangential component of the net force is $$mgsin\theta$$ which provides the required tangential acceleration (or, restoring torque for oscillation)This is along the motion of the pendulum. This is zero only when the string is vertical.

So now you can see, at the extreme position, only tangential component of force is present.

You can use mechanical energy conservation for further analysis of pendulum system.

• Thanks! I don't understand the first formula. To get the radial component, I need to know the value of tension. But tension varies with motion. So how do I calculate the radial component of net force? – Lenny White May 19 at 17:19

net force = t mg sin theta as im on my mobile i wont be able to explain u clearly but if draw an imaginary line going up from the bob u will see that it will make a theta angle with tension .(alternate interior angle) which implies tcos theta =mg and the force which provide acceleration tension *sin theta

• what does t stand for? – Lenny White May 19 at 10:25
• it stands for tension – Bhavay May 19 at 10:25
• but how do I calculate for t? – Lenny White May 19 at 10:32
• t=mg/costheta as net force in vertical.direction is 0 for horizonatal direction u can subsitute value of tension which will provide accleration that is tsintheta=ma – Bhavay May 19 at 10:33
• so net force = mg/cos(θ) * mgsin(θ)? – Lenny White May 19 at 10:36

A is simply A=-mgsin(theta)

The tension in the string has no effect on the restoring force, which is dependent only on the mass and the angle to the vertical.

• Thanks! What about its (net force) direction though? How would I derive that? – Lenny White May 19 at 14:04
• The net force of what? – Physics May 19 at 14:07
• Force A in the clip I linked. I need to know the direction of the net force at a given time. – Lenny White May 19 at 14:08
• Force A is the net force in that direction. Its direction always acts perpendicular to the tension in the string. – Physics May 19 at 14:16
• But it's not always perpendicular if you look at the clip. The direction of force A constantly changes. How do I calculate the direction of force A at given time? – Lenny White May 19 at 14:32