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

Template:80%done The force required to sustain uniform circular motion is $F=ma=m{v}^2/r$, 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?

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

• 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?

Template:Cot $a=\frac{v^2}{r}=\frac{2^2}{2}=2$ Template:Cob

• 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?

Template:Cot $a=\frac{v^2}{r}=\frac{3^2}{3}=\frac{9}{3}=3$ Template:Cob

Proportional reasoning

The force required to sustain uniform circular motion is $F=ma=m{v}^2/r$, 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?

Template:Cot $F=\frac{m{v}^2}{r}\to F=k{v}^2$

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?

Template:Cot $v=\frac{distance}{time}=\frac{2\pi r}{T}\to v=k\frac{r}{T}$

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 Template:Nowrap begin2π = 1Template:Nowrap end? 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 Template:Nowrap consider what happens if the speed is seven meters per blink. OR DO I HAVE IT BACKWARDS????