Circular motion and Gravitation

Question 1

What is formula of angular displacement, θ?

Answer 1

θ = s/r

Question 2

What is one radian?

Answer 2

defined the angle (subtended) at the centre of a circle by an arc equal to the radius of the circle. ∙ For one complete revolution, an object rotates through 360 degrees, or 2π rad.

Question 3

What is angular velocity, ω? What is the formula?

Answer 3

defined as the rate of change of angular displacement

ω = 2π/T = θ/t

Question 4

What is linear velocity? What is the formula?

Answer 4

is the instantaneous velocity at any point in its circular path.
v = r ω

Question 5

What happens to velocity of a body moving in a circle at a constant speed?

Answer 5

The velocity of a body moving in a circle at a constant speed changes since its direction changes. It thus experiences an acceleration, a net force and a change in momentum.

Question 6

What is the formula of centripetal acceleration?

Answer 6

a = r ω2 = v2/r

Question 7

What is centripetal force? What is the formula?

Answer 7

refers to the resultant of all the forces that act on a system in circular motion. {It is not a particular force; “centripetal” simply means “centre-seeking”. Also, when asked to draw a diagram showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as “centripetal force” as that amounts to ‘double-counting’. }
F = m r ω2 = mv2/r

Question 8

Explain weightlessness?

Answer 8

Since the person and the satellite would both have the same acceleration; hence the normal reaction on the person is zero.
{To elaborate: the sensation of weight is due to the normal reaction exerted on the object. When the person & the floor of the satellite have the same acceleration, the contact force between them is zero, hence the normal reaction is zero. This is the state of “weightlessness”.}

Question 9

What is the Newton’s law of gravitation? What is the formula?

Answer 9

The gravitational force of attraction F between two point masses M and m separated by a distance r is given by
F = GMm/r2

Question 10

What is gravitational field strength?

Answer 10

at a point is defined as the gravitational force per unit mass at that point. {NOT: “force on unit mass”} g =F/m
Unit: N kg-1. Vector quantity.

Question 11

What is the formula of field strength due to a point mass M?

Answer 11

g = GM/r2

Question 12

Why is apparent weight is smaller at the poles?

Answer 12

Resultant of the gravitational force and the normal reaction Nequator, provides the centripetal force to keep the body in a circular motion.
GMm/r2 - Nequator = mRω2

Question 13

Explain why is the acceleration of free fall near the earth’s surface approx. constant?

Answer 13

use g = GM/r2
where r = radius of Earth + height above surface (h)
h &laquo_space;RE , hence h is negligible

Question 14

What is gravitational potential and what is the formula?

Answer 14

at a point is defined as the work done per unit mass (by an external force) in bringing a small test mass from infinity to that point (without changing its KE). {Not: a unit mass!}
φ =W/m

Question 15

Why are gravitational potential values negative?

Answer 15

∙ Potential of any point at infinity is zero.
∙ As the gravitational force is attractive, the force exerted by the external agent is opposite in direction to the displacement of the body (in bringing unit mass from infinity to any point in the field), hence work done is negative
∙ The potential, which is the work done per unit mass, are always negative.

Question 16

What is the formula of gravitational potential due to a point mass M?

Answer 16

φ = - GM/r

{The negative sign must not be omitted}

Question 17

What is the r/s between field strength,g and potential,φ?

Answer 17

g = - dφ/dr = - gradient of φ-r graph

Question 18

What is the gravitational potential energy U? What is the formula?

Answer 18

Gravitational potential energy U of any mass m at a point in the gravitational field of another mass M, is the work done in bringing that mass m
{NOT: “unit mass”, or “a mass”} from infinity to that point.
U = m φ = - GMm/r

Question 19

What is the formula for change in GPE?

Answer 19

ΔU = m g h

{only where g is constant over the distance h; otherwise, must use: ΔU = mφf − mφI }

Question 20

What is the formula for total energy of a satellite? How to derive?

Answer 20

Total Energy of a Satellite = GPE + KE = (- GMm/r) + (1/2GMm/r)
= -1/2GMm/r

Question 21

What is the formula for escape speed? How to derive?

Answer 21

Escape speed v = √2gR = √2GM/R
{Need to show by PCE & definition of escape speed; recall & use is not allowed
By Conservation of Energy,
Initial KE + Initial GPE = Final KE + Final GPE
1⁄2 mvE2 + (- GMm/r) = 0 + 0

Question 22

How is period T related to r?

Answer 22

T2 ∝ r3

Question 23

Why does a satellite not move in the direction of the gravitational force?

Answer 23

because the gravitational force exerted by the earth on the satellite is JUST sufficient to cause the centripetal acceleration

Question 24

Why must a geostationary orbit lie in the equatorial plane of the earth?

Answer 24

The force of attraction to the Earth is to its centre so the circular orbit must be centred on the Earth’s centre.
∙ The plane of the orbit must lie in the equatorial plane of the Earth; otherwise it would have a varying latitude (and thus not be geostationary). {N2003 P3Q2 c}