In Episode 20, we are hanging around with a simple pendulum, and discussing just what makes a clock tick.
In Episode 20, we are hanging around with a simple pendulum, and discussing just what makes a clock tick (0:36). While introducing the simple pendulum (1:04), we look at an experiment you can easily do from your own home with a string and a mass (1:33). What factors affect the motion of a pendulum (2:46)? You can test each one at home (4:42), but we let you in on all of the secrets eventually. Finally, conservation of energy has a cameo and we talk about find the velocity of a pendulum (7:01).
The Question of the Day asks: (7:47)
How long would a pendulum arm need to be to complete a tick or a tock every 1 second at sea level on Earth?
Thank you for listening to The APsolute RecAP: Physics 1 Edition!
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on the simple pendulum and the simple harmonic motion that it exhibits.
Let’s Zoom out:
Unit 6 – Simple Harmonic Motion
Topic 6.1-6.2
Big Idea – Force Interactions and Conservation
Tick… Tock… Tick… Tock… ever wonder why grandfather clocks have a large mass that swings back and forth over and over and over? It should be no surprise that the motion of that mass is explained using physics.
Let’s Zoom in:
The large mass that swings back and forth in some clocks is known as a pendulum, and the way it swings is very predictable and consistent. Their motion is so consistent in fact that we can keep time with it. We call this predictable swinging - simple harmonic motion, because it is so repeatable that if we graphed the left and right displacement of one of these pendula over time - it would be sinusoidal! (like a sine wave).
The beauty of a pendulum is that it can easily be replicated at home. Grab a string, some twine, some thread, heck… a shoelace! On one end, tie the string to a fixed location, maybe above a door frame. On the other end attach a mass. It doesn’t need to be super heave, but the thicker the string you used, the heavier the mass you will want. If you chose to use a shoelace, this can even be the shoe it's still attached to! Now, displace the mass slightly from its resting location, a point we will refer to as the equilibrium position, and let it go. Time it, maybe use your cellphone’s stopwatch feature. You are aiming to time one complete oscillation, or back and forth. If it starts on the right, let it swing left, and come all the way back to where it started. This time for one oscillation is known as the pendulum’s period and is measured in seconds. Alternatively, you could count how many back and forth cycles the pendulum makes every second. That is known as the pendulum’s frequency, and is measured in cycles per second or Hertz. Frequency and period are inverse concepts, so taking the inverse of one, will give you the other.
Now it’s time to do some science! What factors do you think might affect the period of the pendulum? There are a few things that come to mind. Maybe we could change the mass of the pendulum. Go ahead, swap it out on your homemade pendulum, and do some science! You will find that whether you increase or decrease the mass, you will see a negligible change to the period of the pendulum. What else could we try? Maybe how far we pull the pendulum back before releasing it? Give it a shot, but be sure you keep the angle of the pendulum string to vertical less than 15 degrees. It is a bit complicated why this must be the case, but essentially, a pendulum does not actually exhibit harmonic motion, but is darn close as long as the angle is less than 15 degrees. ANy changes to period when you alter the release angle? Nope. Ok, how about if we change the length of the pendulum string? Any changes there? EUREKA! We have done it!
It turns out that increasing the length of the string, or the pendulum arm, increases the period of the pendulum. If you wanted to, you could certainly determine by how much the period increases for a given length increase. We could try doubling the pendulum arm, and then tripling, and quadrupling, etc. If we did so, we would see that period increases by the square root of the length increase. On your equation sheet you will notice that the period of a pendulum (Capital T) is equal to 2*Pi*the square root of the pendulum arm length over the gravitational acceleration of the planet you are on. So, wait… gravity matters? Oooook cool.
What would the free-body diagram look like for a pendulum as it swings? Well, to determine that, it helps to look at the extremes of a pendulum. If it were held horizontal, the FBD would have a downward arrow only for the gravitational force, but then the string starts to apply more and more tension and maxes out as the mass reaches the bottom of the arc. Gravity is still there, but because the pendulum is moving in a curve, there must be a centripetal force directed towards the center of the curvature, and therefore more upward tension than downward gravitational force. Those same two forces act on the pendulum mass at all places in between the two extremes, with the tension constantly changing throughout the swing and the force of gravity remaining constant. If you tilt the axes of your coordinate system to allow tension to fall along the y-axis, then you can see that there is an x and y component of the gravitational force. The x-component is the one responsible for speeding up or slowing down the pendulum along its curved path. This is known as the restoring force of the pendulum. It is the restoring force that we could use to find the linear or tangential acceleration of our pendulum at any given moment. BE CAREFUL, you cannot use the kinematic equations here since the acceleration is not uniform, but is in fact constantly changing. So, how can we know how fast the pendulum is moving?
Well… to find a pendulum’s velocity, the easiest way is to use conservation of energy concepts and to set the gravitational potential energy at the highest point equal to the kinetic energy of the pendulum bob (another name for the mass at the end of the string) at the lowest point.
To Recap…
The length of the pendulum arm and the gravitational acceleration at the location where the pendulum is swinging both affect the period of the pendulum. The restoring force for a pendulum is equal to m*g*sine(theta), and to find the speed or height of a pendulum bob at any location, use conservation of energy.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at the conical pendulum, and the factors that affect how quickly it oscillates.
Today’s Question of the Day focuses on the simple pendulum.
Question: How long would a pendulum arm need to be to complete a tick or a tock every 1 second at sea level on Earth?