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

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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?

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

• 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?

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

• 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?

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

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?

click for answer
$F={\frac {mv^{2}}{r}}\rightarrow F=kv^{2}$ 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?

click for answer

$v={\frac {distance}{time}}={\frac {2\pi r}{T}}\rightarrow v=k{\frac {r}{T}}$ 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 2π = 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 2π blinks consider what happens if the speed is seven meters per blink. OR DO I HAVE IT BACKWARDS????