How things work college course/Conceptual physics wikiquizzes/Uniform circular motion

The force required to sustain uniform circular motion is $$F=ma=mv^2/r$$, where $$a$$ is acceleration and $$r$$ is the radius of the circle. The period of orbit, $$T$$, is related to velocity $$v$$ by the fact that the distance traveled in one period of orbit is the circumference of the circle, $$2\pi r$$.

You will be given the aforementioned equations as you are asked to do the following problems:

Plug in the numbers

 * What is the acceleration (in m/s2) of a particle that is traveling on a circle with a radius of 2 meters at a speed of 3 meters per second?

$$a=\frac{v^2}{r} = \frac{3^2}{2}=\frac{9}{2}=\frac{8}{2} + \frac 1 2 = 4.5$$


 * What is the acceleration (in m/s2)of a particle that is traveling on a circle with a radius of 2 meters at a speed of 2 meters per second?

$$a=\frac{v^2}{r} = \frac{2^2}{2}=2$$


 * What is the acceleration (in m/s2)of a particle that is traveling on a circle with a radius of 3 meters at a speed of 3 meters per second?

$$a=\frac{v^2}{r} = \frac{3^2}{3}=\frac{9}{3}=3$$

Proportional reasoning
The force required to sustain uniform circular motion is $$F=ma=mv^2/r$$, where $$a$$ is acceleration and $$r$$ is the radius of the circle. The period of orbit, $$T$$, is related to velocity $$v$$ by the fact that the distance traveled in one period of orbit is the circumference of the circle, $$2\pi r$$.

Mr. Smith is using a string to swing a rock so fast that gravity may be neglected. What happens to the tension in the string when the velocity doubles?

$$F=\frac{mv^2}{r} \rightarrow F=kv^2$$

where k is a constant that depends on units. Adopting units such that F=v=1 initially, we see that k=1. In these units the force after changing v equals 2. The final force in these units is

$$F=v^2=2^2=4$$ or 4 times the initial force.

Answer: The force increases by a factor of 4 when the speed is doubled.

Mr. Smith is using a string to swing a rock so fast that gravity may be neglected. What happens to the speed of the rock if the period is cut in half while the radius is tripled?

$$v=\frac{distance}{time}=\frac{2\pi r}{T} \rightarrow v=k\frac r T$$

where k is a constant that depends on units. Adopting units such that v=r=T initially, we see that k=1. In these units, the final radius is 3 and final period is 1/2. The final speed in these units is

$$v=\frac r T = \frac{3}{\frac 1 2}= \frac {3\cdot 2} {\frac 1 2 \cdot 2} = 6$$

Answer: The speed is increases by a factor of 6 when the the radius is tripled and the period is cut in half.

Aside: If k=1 am I saying that 2&pi; = 1? I hope not! Just measure velocity in meters per second, and measure time a made-up unit we shall call a blink. If one second equals 2&pi; blinks consider what happens if the speed is seven meters per blink. OR DO I HAVE IT BACKWARDS????