Motion in a circle

Curvilinear motion, uniform motion in a circle: period, frequency, linear and angular velocity, centripetal acceleration.

Terms

Studied 0 of 7

Formulas

Period and frequency

T = \dfrac{t}{N}, \qquad \nu = \dfrac{N}{t} = \dfrac{1}{T}

Quantities
Symbol	Name	Unit
$T$	period	s
$\nu$	frequency	Hz
$N$	number of turns	—
$t$	time	s

Solved for another quantity:

T = \dfrac{1}{\nu}

N = \nu\,t

Linear speed

v = \dfrac{2\pi R}{T}

Quantities
Symbol	Name	Unit
$v$	linear speed	m/s
$R$	radius of the circle	m
$T$	period	s

Solved for another quantity:

v = 2\pi R\,\nu

T = \dfrac{2\pi R}{v}

Note. In one period the body covers the circumference $2\pi R$ .

Angular velocity

\omega = \dfrac{\varphi}{t} = \dfrac{2\pi}{T}

Quantities
Symbol	Name	Unit
$\omega$	angular velocity	rad/s
$\varphi$	angle turned by the radius	rad
$t$	time	s
$T$	period	s

Solved for another quantity:

\omega = 2\pi\nu

Linear and angular velocity

v = \omega R

Quantities
Symbol	Name	Unit
$v$	linear speed	m/s
$\omega$	angular velocity	rad/s
$R$	radius	m

Solved for another quantity:

\omega = \dfrac{v}{R}

Note. A point on the rim of a disk moves faster than a point near the centre, although their angular velocity is the same.

Centripetal acceleration

a_{\text{ц}} = \dfrac{v^2}{R}

Quantities
Symbol	Name	Unit
$a_{\text{ц}}$	centripetal acceleration	m/s²
$v$	linear speed	m/s
$R$	radius of the circle	m

Solved for another quantity:

a_{\text{ц}} = \omega^2 R

v = \sqrt{a_{\text{ц}} R}

Note. Speed enters squared: at twice the speed on the same radius the acceleration is four times as large.

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Motion of the body

The velocity is tangent to the circle, the acceleration points to the centre. The dots mark the body's position every 0.5 s.

t

= 0 sPlayback speed

Graphs

The graphs are drawn as the body moves. Click a graph to move through the recording.

l(t)

Distance grows steadily: $2\pi R$ per turn.

v(t)

The speed is constant.

a(t)

The acceleration has a constant magnitude and points to the centre.

t

0 s

N

l

0 m

s

0 m

Formulas with your numbers

v = \dfrac{2\pi R}{T} = \dfrac{2\pi\cdot 3}{4} = 4.71\ \text{m/s}

a_{\text{ц}} = \dfrac{v^2}{R} = \dfrac{4.71^{2}}{3} = 7.39\ \text{m/s²}

\nu = \dfrac{1}{T} = \dfrac{1}{4} = 0.25\ \text{Hz}

\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{4} = 1.57\ \text{rad/s}

N = \dfrac{t}{T} = \dfrac{0}{4} = 0

How to read the graphs

The speed $v(t)$ does not change — it is a horizontal line. But the velocity vector keeps turning, so its component $v_x(t)$ swings between $-v$ and $v$ .
Distance $l(t)$ grows steadily: by the circumference $2\pi R$ in every period $T$ . The displacement after a full turn is zero.
The acceleration $a(t)$ has a constant magnitude and always points to the centre. It changes the direction of the velocity, not its magnitude.
All points of a rotating disk share the same $T$ and $\omega$ , while $v = \omega r$ and $a = \omega^2 r$ are proportional to the distance from the centre: the $v(t)$ and $a(t)$ graphs follow the shape of $r(t)$ .

Loading…

Terms

Studied 0 of 7

Formulas

Period and frequency

T = \dfrac{t}{N}, \qquad \nu = \dfrac{N}{t} = \dfrac{1}{T}

Quantities
Symbol	Name	Unit
$T$	period	s
$\nu$	frequency	Hz
$N$	number of turns	—
$t$	time	s

Solved for another quantity:

T = \dfrac{1}{\nu}

N = \nu\,t

Linear speed

v = \dfrac{2\pi R}{T}

Quantities
Symbol	Name	Unit
$v$	linear speed	m/s
$R$	radius of the circle	m
$T$	period	s

Solved for another quantity:

v = 2\pi R\,\nu

T = \dfrac{2\pi R}{v}

Note. In one period the body covers the circumference $2\pi R$ .

Angular velocity

\omega = \dfrac{\varphi}{t} = \dfrac{2\pi}{T}

Quantities
Symbol	Name	Unit
$\omega$	angular velocity	rad/s
$\varphi$	angle turned by the radius	rad
$t$	time	s
$T$	period	s

Solved for another quantity:

\omega = 2\pi\nu

Linear and angular velocity

v = \omega R

Quantities
Symbol	Name	Unit
$v$	linear speed	m/s
$\omega$	angular velocity	rad/s
$R$	radius	m

Solved for another quantity:

\omega = \dfrac{v}{R}

Note. A point on the rim of a disk moves faster than a point near the centre, although their angular velocity is the same.

Centripetal acceleration

a_{\text{ц}} = \dfrac{v^2}{R}

Quantities
Symbol	Name	Unit
$a_{\text{ц}}$	centripetal acceleration	m/s²
$v$	linear speed	m/s
$R$	radius of the circle	m

Solved for another quantity:

a_{\text{ц}} = \omega^2 R

v = \sqrt{a_{\text{ц}} R}

Note. Speed enters squared: at twice the speed on the same radius the acceleration is four times as large.

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Motion of the body

The velocity is tangent to the circle, the acceleration points to the centre. The dots mark the body's position every 0.5 s.

t

= 0 sPlayback speed

Graphs

The graphs are drawn as the body moves. Click a graph to move through the recording.

l(t)

Distance grows steadily: $2\pi R$ per turn.

v(t)

The speed is constant.

a(t)

The acceleration has a constant magnitude and points to the centre.

t

0 s

N

l

0 m

s

0 m

Formulas with your numbers

v = \dfrac{2\pi R}{T} = \dfrac{2\pi\cdot 3}{4} = 4.71\ \text{m/s}

a_{\text{ц}} = \dfrac{v^2}{R} = \dfrac{4.71^{2}}{3} = 7.39\ \text{m/s²}

\nu = \dfrac{1}{T} = \dfrac{1}{4} = 0.25\ \text{Hz}

\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{4} = 1.57\ \text{rad/s}

N = \dfrac{t}{T} = \dfrac{0}{4} = 0

How to read the graphs

The speed

v(t)

does not change — it is a horizontal line. But the velocity vector keeps turning, so its component

v_x(t)

swings between

-v

and

v

Distance

l(t)

grows steadily: by the circumference

2\pi R

in every period

T

. The displacement after a full turn is zero.

The acceleration

a(t)

has a constant magnitude and always points to the centre. It changes the direction of the velocity, not its magnitude.

All points of a rotating disk share the same

T

and

\omega

, while

v = \omega r

and

a = \omega^2 r

are proportional to the distance from the centre: the

v(t)

and

a(t)

graphs follow the shape of

r(t)