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

 Completion status: this resource is ~80% complete.

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

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?

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

• 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?

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

• 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?

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

## Proportional reasoning

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

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?

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

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

${\displaystyle F=v^{2}=2^{2}=4}$ 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?

${\displaystyle v={\frac {distance}{time}}={\frac {2\pi r}{T}}\rightarrow v=k{\frac {r}{T}}}$
${\displaystyle v={\frac {r}{T}}={\frac {3}{\frac {1}{2}}}={\frac {3\cdot 2}{{\frac {1}{2}}\cdot 2}}=6}$