# Pendulum issues - find acceleration and restoring force

• UrbanXrisis
In summary, the max angular acceleration for a simple pendulum with a mass of 0.25kg, length of 1m, and displaced 15 degrees is 0.668rad/s^2. The incorrect calculation of 0.817m/s for the maximum velocity is due to using the linear approximation instead of the exact equation of motion. The maximum angular velocity can be found directly from the exact equation or the linearized version, but it does not necessarily occur at the same point as the maximum acceleration.
UrbanXrisis
I am given that the mass of a simple pendulum is 0.25kg, length 1m and displaced 15 degrees then released.

a) max angular acceleration?
b) max restoring force?

for a... I vound that the max velocity is 0.817m/s

$$\omega=\frac{v}{r}$$
$$\omega=\frac{0.817m/s}{1m}$$
$$\omega=0.817rad/s$$

$$\alpha=r \omega ^2$$
$$\alpha=1m * 0.817^2$$
$$\alpha=0.668rad/s^2$$

I know that's wrong but why?

for b...
I just multiply alpha by the radius to get acceleration then multiply that by the mass.

It's hard to tell what you are doing wrong because it's not clear what you did.

You say you got "max velocity is 0.817m/s". How? Are you using the exact equation for the motion or the linear approximation.

It seems strange to me that you would find the linear velocity, v, when the problem asks for the angular velocity and it is more direct to find it directly.

The exact equation of motion, F= ma, for a pendulum is $$ml\frac{d^2\theta}{dt^2}= -mg sin(\theta)$$. The "m"s cancel so we have $$\frac{d^2\theta}{dt^2}= -\frac{g}{l}sin(\theta)$$. Let $$\omega= \frac{d\theta}{dt}$$ (the angular velocity). By the chain rule $$\frac{d^2\theta}{dt^2}= \frac{d\omega}{dt}= \frac{d\omega}{d\theta}\frac{d\theta}{dt}= \omega\frac{d\omega}{d\theta}$$ so the equation becomes $$\omega\frac{d\omega}{d\theta}= -\frac{g}{l}sin \theta$$.

That's a separable equation: $$\omega d\omega= -\frac{g}{l}sin \theta d\theta$$ which integrates to $$\frac{1}{2}\omega^2= \frac{g}{l}cos \theta$$.
You should be able to find the maximum value of $$\omega$$ directly from that.

The "linearized" version of that is $$l\frac{d^2\theta}{dt}= -g \theta$$ (since , for small $$\theta$$, $$sin \theta$$ is approximately $$\theta$$) which is the "linear homogeneous d.e. with constant coefficients", $$\frac{d^2\theta}{dt^2}+ \frac{g}{l}\theta= 0$$. The general solution for that is $$\theta= Ccos(\sqrt{\frac{g}{l}}t)+ Dsin(\sqrt{\frac{g}{l}}t)$$. The angular velocity is $$\omega= \frac{d\theta}{dt}= (\sqrt{\frac{g}{l}})(-Csin(\sqrt{\frac{g}{l}}t)+ Dcos(\sqrt{\frac{g}{l}}t))$$. Again, you can find the maximum angular velocity directly from that.

I think he used conservation of energy.At zero PE (the lowest point of the circular trajectory),the KE is maximum and -------->max.linear velocity...

My guess.

Daniel.

UrbanXrisis said:
I am given that the mass of a simple pendulum is 0.25kg, length 1m and displaced 15 degrees then released.

a) max angular acceleration?
b) max restoring force?

for a... I vound that the max velocity is 0.817m/s

$$\omega=\frac{v}{r}$$
$$\omega=\frac{0.817m/s}{1m}$$
$$\omega=0.817rad/s$$

$$\alpha=r \omega ^2$$
$$\alpha=1m * 0.817^2$$
$$\alpha=0.668rad/s^2$$

I know that's wrong but why?

for b...
I just multiply alpha by the radius to get acceleration then multiply that by the mass.
Maximum velocity doesn't have to occur at the same point as your maximum acceleration. HallsofIvy gives the mathematical explanation, but you should be able to see this visually, as well. Gravity is providing the force that accelerates the pendulum, but the pendulum is restrained by the rod holding it to the center. Your maximum acceleration will come when the pendulum's motion is straight down ; when the motion is aligned with the force of gravity (or when it's as close to straight down as it's ever going to get). For a pendulum, your acceleration will hit it's maximum velocity when it's at the bottom of its swing, a point where the force of gravity is perpendicular to the pendulum's motion, meaning acceleration will be zero.

You should be able to see where your maximum acceleration has to occur and figure it out from there.

## 1. What is a pendulum?

A pendulum is a weight suspended from a fixed point that is free to swing back and forth due to the force of gravity. It consists of a mass (bob) attached to a string or rod, and its motion follows the laws of simple harmonic motion.

## 2. How do you find the acceleration of a pendulum?

The acceleration of a pendulum can be calculated using the formula a = -(g/l)sin(Θ), where g is the acceleration due to gravity (9.8 m/s^2), l is the length of the pendulum, and Θ is the angle the pendulum makes with the vertical. This formula assumes that the amplitude of the pendulum's swing is small (<15°).

## 3. What is restoring force in a pendulum?

Restoring force in a pendulum refers to the force that brings the pendulum back to its equilibrium position. In the case of a simple pendulum, this force is provided by gravity as the pendulum bob moves towards its lowest point, causing an increase in tension in the string or rod.

## 4. How does the length of a pendulum affect its motion?

The length of a pendulum directly affects its period, or the time it takes for one complete swing. A longer pendulum will have a longer period, meaning it will take longer to complete one swing. This is because the longer length increases the distance the pendulum bob must travel, resulting in a slower swing.

## 5. Can you use a pendulum to measure the strength of gravity?

Yes, a pendulum can be used to measure the strength of gravity by measuring the period of the pendulum's swing. The formula for the period of a pendulum is T = 2π√(l/g), where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity. By measuring the period and knowing the length of the pendulum, the value of g can be calculated.

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