Question

If an object moves in uniform circular motion in a circle of radius R = 1.0 meter, and the object takes 4.0 seconds to complete ten revolutions, calculate the centripetal acceleration.

Answer 1

Answer:

$a_c=246.5 \frac{m}{s^2}$

Explanation:

The formula for centripetal acceleration is:

$a_c=rw^2$

Where,

$r:$ radius.

$w:$Angular speed.

Angular speed is defined by:

$w=\frac{\Delta\theta}{\Delta t}$

Where,

$\theta:$ position angle.

$t:$ time.

In this case we have that the object takes 4.0 seconds to complete ten revolutions.

You have to know that 1 revolution = $2\pi rad$, then

10 revolutions= $10.2\pi rad = 20\pi rad$

Replacing

$\Delta\theta=20\pi rad\\\Delta t=4.0s$

in the formula of Angular speed:

$w=\frac{\Delta\theta}{\Delta t}\\\\w=\frac{20\pi rad}{4.0s}$

$w=5\pi \frac{rad}{s}\\w=15.7\frac{rad}{s}$

Now we have,

$a_c=rw^2\\a_c=r(15.7\frac{rad}{s})^2$

r=1.0m

$a_c=1.0m(15.7\frac{rad}{s})^2\\\\a_c=1.0(246.5)\frac{m}{s^2} \\\\a_c=246.5\frac{m}{s^2}$

Then the centripetal acceleration is:

$a_c=246.5 \frac{m}{s^2}$

Answer 2

A = ω²r

r = 1 m
ω = 2πf = 2π * 10 / 4 = π*5 s⁻¹