How come the net force of the pendulum (simple harmonic motion) is changing but an object with uniform rotational motion does not?

Question

a sphere is attached to an end of the cord, and it is raised to a certain angle above its equilibrium (lowest) point on the left, and then is released. At the point it was being released, the tension force is at the minimum, mg always stays constant when the sphere is doing its harmonic motion, and then, when it reaches the equilibrium (lowest) point, tension force gets to its maximum magnitude which is equals to mg where the acceleration becomes zero. As the sphere keep moving to the right, it will eventually stopped at the point where it was about to release. During this simple harmonic motion, the net force is changing so as the magnitude of acceleration.(?) This change in force is due to the angle of the string is changing as it is swings back and forth, therefore cause the tension changes.(?)

However, a object that is in uniform rotational motion, its speed is constant while its direction changes all the time, but the net force stays the same.

so my question is, why does the net force of a sphere in the swinging pendulum is changing but it does not change in a uniform rotational motion since the string in both cases all have different angles at every moment, therefore their force of tension should be changing, isn't it ?

Could you please point out what's wrong in my thoughts and statements ( especially the one l have a question mark after the sentence )?

_Second question

According to the conservation of energy (?), the ball will reach the same amplitude on the right hand side, assume that the air resistance is negligible, then, the sphere in the first sketch from left to right will move the right along an arc line, but what about the second and the third sketches, will they also move along an arc line reaches their original position at the other side ? why? how does this different from an uniform rotational motion ?

Answer 1

SHM occurs only in special cases. In case of simple pendulum it occurs when angular displacement is small. If you have done the derivation you must have come through it . After that you can yourself check the net force using equations of SHM . Now as far as uniform rotational motion is considered it's net force is constant in magnitude because for every body to be circular motion centripetal force is required (which is the force you are talking about ). This force acts perpendicular to the velocity thus it does not change magnitude of velocity but only direction. Also only the magnitude of force is constant in this case. Now coming to the diagrams , the first one is SHM(if displacement is small) and if not still it would come to its original position as you mentioned. Se is the case for second diagram . But in the third one if it is connected by a string then it would simply fall down(if you do not provide the ball with a horizontal velocity sufficient for it to complete a vertical circle.) . But if it is a rod then it would also move through a complete circular path and come to its original position (neglecting friction and air resistance).
You should not confuse between an SHM , uniform rotational motion and motion in a vertical circle. SHM occurs only in some special cases, rotational motion can occur anywhere provided there is a force perpendicular to velocity and motion in a vertical circle also has some constraints.

Vertical Circular Motion

Answer 2

First of all, a bit of clarification:

1. The force (and therefore the acceleration) in a circular uniform motion is not constant, it changes in direction. However, it is true that, is the angular velocity $\omega=v/L$ is constant, the force's magnitude is constant. In the case of a pendulum, the force changes both magnitude and direction over time.
2. The two motions (pendulum and circular) are both periodic but they are different in general.

Second of all: I wil call $\alpha$ and $\omega$ the angular acceleration and speed, $v$ the tangential speed and $L$ the length of the pendulum / radius of the circular motion.

Having set this, now the main difference between the two motions is that in a harmonic motion you have a constant change of energy from kinetic to potential, i.e. kinetic energy turns into potential energy (be it gravitational energy in a pendulum of elastic energy in a spring) and then in turn potential energy changes into kinetic energy. This is because there is an external (conservative) force which is doing some work on the system. In both the case of a pendulum and of circular motion, the tension of the string (or whatever centripetal force you are applying in circular motion) are not doing any work because they are always perpendicular to the displacement of the mass. Their only role is changing the direction of the velocity keeping it tangential at all times. In the case of a pendulum, however, you have gravity which is taking some of the kinetic energy and thus decelerating or accelerating the mass locally. Hence the change in speed therefore the acceleration therefore the change in force felt by the mass.

In the case of circular motion on the other hand, as there is in general no external force, the kinetic energy hence the speed hence the acceleration hence the force are (in magnitude) constant.

So for a circular motion you have, as equation, $$\alpha = 0$$ $$\omega = v/L = const $$ so the equation for the angular acceleration reads

$$\alpha = d^2\theta/dt^2 = 0$$

So no external force / torque ($\alpha = I\tau = 0$ with $I$ moment of inertia of your mass - i.e. Newton's second law for rotations).

On the other hand, for a pendulum, you have an harmonic motion $$\alpha = d^2\theta/dt^2 =-{g\over L}\theta$$ so you have an angle-dependent torque $\tau =-{g\over L}\theta$ that causes the pendulum to change speed. And it comes from gravity which "slows down / accelerates" the mass as it changes position differently (because the tension "steals" some of the gravity).

If you take a pendulum and put it on a frictionless horizontale plane (i.e. no gravity) and you give it a spin it will keep spinning at constant angular velocity as in circular motion.

So the motions are in general different. However, there are analogies i.e. circular motion and pendulum can be turned into one-another by adding/removing an external force.

If you take a circular motion and flip it so that now gravity is acting on it, then the speed (hence the acceleration and the force) will change over time as gravity acts on the mass as an external force, "stealing" some of the kinetic energy (which it later gives back). If your initial motion was "small" so that a small-angle approximation is valide, then your circular motion turns into a pendulum just by the addition of a restoring force (gravity). Otherwise, it is a more complex motion given by the equation for the angle

$$\alpha = d^2\theta/dt^2 = -{g\over L} sin(\theta)$$ (which becomes the pendulum equation if $\theta\approx0$ and $sin(\theta)\approx\theta$) which has a angle-dependent force - so the force will change over time - but you see already that if you have $g=0$ (i.e. no gravity) then $$d^2\theta/dt^2 = 0$$ i.e. you recover a uniform motion given by $\omega=const$.

Here a small GIFs showing you the difference between the two motions: they are exactly the same, they start the same position and speed BUT one has gravity acting on it and the other one does not. So while they are both periodic (with same period) one oscillates (because of gravity pushing it back) and the other one does not. You can think of one being on plane, one being on plane with gravity.

Answer 3

so my question is, why does the net force of a sphere in the swinging pendulum is changing but it does not change in a uniform rotational motion since the string in both cases all have different angles at every moment, therefore their force of tension should be changing, isn't it ?

In the case of uniform circular motion, the (magnitude of) net force is only constant if the provided external force makes it so! In other words: $$F_{net}(\theta)=F_{ext}(\theta)-\left( mg-T(\theta)\right)$$

According to the conservation of energy (?), the ball will reach the same amplitude on the right hand side, assume that the air resistance is negligible, then, the sphere in the first sketch from left to right will move the right along an arc line, but what about the second and the third sketches, will they also move along an arc line reaches their original position at the other side ? why? how does this different from an uniform rotational motion ?

The difference between this motion (which is no longer SHM) and uniform rotational motion is that the speed is not constant here. This is because like it was mentioned prior, the net force has to be constant for uniform rotation. But here the net force is a function of the angle. So this motion is a more complicated one. As you can see below. The speed is changing with the angle.