Uniform circular motion

Links and useful resources

Lightning Round Questions

What is Newton's first law of motion? (inertia)
What is Newton's second law of motion? (f=ma)
What is Newton's third law of motion? (equal/opposite reaction)
How much gravitational potential energy does a 10 kg mass have if it is 20 meters above the ground?
What is the formula for calculating work?
Explain the idea of power in physics.

gr7: [lightning:: 2]
gr10: [lightning:: 4]

Demonstration

Attach a spring to the shaft of the drill and spin with a weight attached.

Measure angular speed
Measure force on spring (distance)
Change speed and repeat for a few speeds.

If the centripetal force is $mr\omega {#2}$ , what would happen to the surface of a bowl of water if you were to spin it at a fixed speed?

Imagine there's a straw full of water, what happens to the pressure in it?
It has the centripetal force for its distance, but it also has to support all of the other water closer to the center!
The pressure goes up as the square of the radius, which means... so does the depth!
The surface of the water forms a perfect parabola! You can even use this to make a liquid mirror telescope.

Concept summary and connections

centripetal acceleration
centripetal force

Lesson content with examples

An object moving in a circle is always accelerating. After all, it's velocity vector is changing, so there must be acceleration happening. This is where we finally break the idea that accelerating = going faster once and for all! You can go in a circle without changing speed, forever. If the planets were it perfectly circular orbits, this is exactly what they would do (for some values of "forever" including "a really, really long time").

So how can that be? How can you accelerate forever without any energy going in to the system? How is it that your velocity can be constantly changing but your speed is constant? The answer to both is from one idea:

In circular motion, the acceleration always points toward the center of the rotation

We know from geometry that that a line from the center of a circle is always perpendicular to the circle where it crosses. We also know that work only happens when some component of the motion is parallel to the force. Taking those together, we can see that the centripetal force doesn't do any work! It's always perpendicular to the object, so it doesn't change its kinetic energy.

The math

Centripetal acceleration is inversely proportional to the radius of the arc, and directly proportional to the squared speed of the motion. The larger the arc, the lower the acceleration. The higher the speed, the higher the acceleration. $a_{c} = \frac{v^{2}}{r}$ . We usually write this using angular velocity though:

$v = ω r$ , so $v^{2} = \frac{ω^{2} r^{2}}{r} = ω^{2} r$

So, we can write $a_{c} = ω^{2} r$ , or centripetal acceleration is the angular velocity squared time the radius of the turn.

We also know that $force = mass \times acceleration$ , so the centripetal force must be $f_{c} = m ω^{2} r$

From that, we can figure out a whole bunch of neat stuff.

Media resources

Guided practice

A tennis ball on a string, spinning. What happens when you double the length of the string?
What happens when you reel in the string so that it's 1/4 the length?
If you double the speed of the tennis ball, how much does the force on the string increase?
A car with mass 1000kg is driving over the top of a circular hill with radius 20m. The car's speed is 100 m/s. Will the car leave the ground? What are the forces? Make a free body diagram and motion diagram, then solve for all forces. Class needs to do this in notebook while I work on the board

Homework

#hw (physics) Brain teaser: IF $work = force \times distance$ , and you tie a tennis ball to a string and spin it in space so it never stops, is the string doing work on the tennis ball? If so, where does the energy come from? 2025-02-19
#hw (physics) Do the centripetal force simulation 2025-02-19
#hw (physics) Do the uniform circular motion concept checker in Sudden Death mode 2025-02-19
#hw (physics) CG5 mission - circular motion 2025-02-28
#hw (physics) Forces and Free body diagrams for circular motion 2025-02-26