Circular Motion & Gravitation

Question 1

An object moving along a circular path at constant speed is still accelerating. Why?

Answer 1

1. Its direction of motion is constantly changing
2. ∴ velocity is constantly changing
3. acceleration is the rate of ∆ velocity

Question 2

a = v^2/r = 4π^2r/T^2

Answer 2

• Only constant speed questions
  a = acceleration (ms-2)
  v = speed (ms-1)
  r = radius of circle (m)
  T = period (time for one revolution)(s)

Question 3

F = mv^2/r

Answer 3

F = centripetal force (N)
m = mass (kg)
r = radius (m)

Question 4

v = 2πr/T

Answer 4

v = speed (ms-1) (tangent to the circle)
r = radius of circle (m)
T = period (time for one revolution)(s)

Question 5

ω = θ/t

Answer 5

ω = angular velocity (rads^-1)
θ = angular displacement (rad)

one revolution = 360° = 2π radians

Question 6

T = 1/f

Answer 6

T = period (time for one revolution)(s)
f = frequency (Hz)

Question 7

ω = 2π f

Answer 7

ω = angular velocity (rads^-1)
f = frequency (Hz)

not given

Question 8

v = ωr

Answer 8

v = linear velocity (ms-1)
ω = angular velocity (rads^-1)
r = radius (m)

Question 9

F = mω^2r

Answer 9

F = centripetal force (N)
m = mass (kg)
ω = angular velocity (rads^-1)
r = radius (m)

Question 10

F = mv^2/r

Answer 10

F = centripetal force (N)
m = mass (kg)
v = linear velocity (ms-1)
r = radius (m)

Question 11

Fc = mgtanθ

Answer 11

Fc = centripetal force (N)
m = mass (kg)
g = gravity (9.81 ms-2)

*ignore friction

Question 12

v = √rgtanθ

Answer 12

• maximum speed for a banked corner of angle θ (if there is no sideways friction)

Question 13

v = √rg

Answer 13

minimum speed needed to move in a vertical circle
- top of the circle (feel weightless) Fc = Fg
Fg = weight force

Question 14

Ek (bottom) = Ek (top) + Ep (top)

Answer 14

Top of circle = some Ep + some Ek
Bottom = all Ek
Top of hill = all Ep

Question 15

Newtons universal law of gravitation states:

Answer 15

“every single point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation”

*point mass = a mass which does not take up any space

Question 16

F = G Mn/r^2

Answer 16

F = gravitational force (N)
M & m = mass (kg)
r = distance/radius (m)
G = gravitational constant = 6.67 x 10^-11 Nm^2kg^-2

Question 17

Gravitational field is defined as :

Answer 17

“A region of space where a mass experences a force because of its mass”

Earth = radial
surface of earth = uniform

Question 18

Gravitational field strength (g) is defined as:

Answer 18

“the force per unit mass experenced by a small point test mass placed in the field”

*A test mass is one that has such a small mass that it does not change the gravitational field in which it is placed

Question 19

g = F/m

a = F/m
*assuming no air resistance

Answer 19

g = Gravitational field strength (Nkg^-1)
F = force (N)
m = mass (kg)

a = acceleration (ms^-2)

Question 20

g = GM/r^2

*Gravitational field strength at a planet’s surface

Answer 20

r = radius (m) (from centre of planet)
g = Gravitational field strength (Nkg^-1)
G = gravitational constant = 6.67 x 10^-11 Nm^2kg^-2
M = mass (kg)

Question 21

v (of satellite) = √GM/r

Answer 21

r = radius (m) (from centre of planet)
G = gravitational constant = 6.67 x 10^-11 Nm^2kg^-2
M = mass (kg)
v = speed of satellite (ms-1)

Centripetal force = Gravitational force

Question 22

a (acceleration of a satellite) = Fg(or Fc)/m
or
a (acceleration of a satellite) = v^2/r

Answer 22

m = mass of satellite
Fg = gravitational force
Fc = centripetal force
v = speed of satellite
r = radius (m) (from centre of planet)

Question 23

r^3 = GMT^2/4π^2

Answer 23

G, M, 4, π are all constants so r^3 ∝ T^2

i.e. r^3 / T^2 = constant (for anything orbiting a body of mass)

Question 24

Angular displacement (θ)

Answer 24

angle (in radians) through which an object undergoing

circular motion has moved

Question 25

angular velocity
(ω)

Answer 25

angular displacement divided by time taken

(units: rad s-1)

Question 26

centripetal force

Answer 26

an unbalanced force that causes circular motion

Question 27

frequency

Answer 27

number of revolutions per sec

Question 28

geostationary orbit

Answer 28

an orbit around the Earth that has a period of 24 hours
(so the satellite remains above the same point on the

Earth’s surface)

Question 29

Kepler’s 3rd Law

Answer 29

the cube of the radius of a satellite is proportional to the square of its period

Question 30

period

Answer 30

time for one revolution

Question 31

point mass

Answer 31

a mass which does not take up any space (i.e. it is infinitely small)

Question 32

test mass

Answer 32

a mass that is so small it does not change the gravitational field in which it is placed