Unit 6: Circular Motion and Gravitation Notes

Circular motion and angular speed

Newton’s law of gravitation

6.1.1 Circular motion and angular speed

Angular displacement: The angle through which an object in circular motion has moved from a
given initial point.
Angular speed (ω): The rate of change of angular displacement

Consider the object in Figure 6.2, which rotates in a circle of radius r in a counter-clockwise
direction, with constant speed v. Let T be the time taken to complete one full revolution. We call T
the period of the motion. Since the speed is constant and the object covers a distance of 2πr in a
time of T seconds, it follows that:

2π𝑟
𝑣= 𝑇

As the object moves around the circle it sweeps out an angle Δθ radians in a time Δt, as shown in
Figure 6.2. We can therefore define the angular speed of the object, denoted by ω, by:

∆θ
ω = ∆𝑡

For a complete revolution, Δθ = 2π and Δt = T, so we also have:

2π
ω = 𝑇
= 2π𝑓

The units of angular speed are radians per second, rad s−1. The velocity vector is at a tangent to
the circle. In a short time Δt the body travels a distance vΔt along the circle. The angle swept in that
same time is Δθ.The distance traveled is an arc of the circle, and from trigonometry we know that
the length of the arc of a circle radius r is given by rΔθ. So we have that:

𝑣∆𝑡 = 𝑟∆θ

Dividing by Δt:

∆θ
𝑣=𝑟 ∆𝑡
𝑣 = 𝑟ω

This is the relation between the linear speed v and the angular speed ω.
6.1.2 Centripetal Acceleration

Acceleration is defined as a change in velocity.

Since velocity is a vector quantity, acceleration can therefore occur when either the magnitude of
velocity (i.e. speed) changes, or the direction of velocity changes.

An object traveling in a circle experiences centripetal acceleration because the direction of its
linear velocity is constantly changing, meaning that the object is accelerating because its
direction is changing, even if its speed is constant.

Centripetal acceleration is given by:

2 2
𝑣 4π𝑟 2
𝑎= 𝑟
= 2 =ω 𝑟
𝑇

The first equation for centripetal acceleration is the most common, and arguably, useful one.

If the object is traveling at a constant speed, the acceleration will always be towards the center of
the circle, tangent to the object’s linear velocity as shown below:

But, if the speed of the object is changing, the real acceleration will not be towards the center of
the circle. However, the centripetal acceleration is taken to be the component of the net
acceleration that is towards the center of the circle.

6.1.3 Centripetal Forces

Due to Newton’s Second Law, for an object to accelerate, the object must experience a net force
in the direction of acceleration. Hence, for an object to travel in a circular path, the object must
experience a net force towards the center of the circle.

This net force is known as the centripetal force, and can be calculated by:
 2
𝑚𝑣 2
𝐹= 𝑟
= 𝑚ω 𝑟

Note that centripetal force is a net force acting on an object, rather than a specific type of force.
This means that centripetal force can be provided by, for example, the gravitational force in the case
of a satellite orbiting a planet, or combined frictional and gravitational force in the case of a car
turning on a banked corner.

For example, consider a car moving on a circular level road of radius r with constant speed v.
Friction between the wheels and the road provides the necessary force directed towards the centre
of the circle that enables the car to take the turn (Figure 6.7). Note that in this example it is
friction that provides the centripetal force. However, this does not mean that friction is always
a centripetal force – it only applies to the case when the resulting motion is circular.

6.1.4 Moving in a vertical circle

When moving in a vertical circle, the gravitational force will impact the net force experienced
by an object in circular motion. For an object on a string, this will mean that the 2 forces acting on
the object are gravity and tension. On the diagram below, the gravitational, tension, and centripetal
force is shown.
Notice how the centripetal force is constant and always directed at the center, the force due to
gravity is constant and always directed towards the ground, but the tension force changes as the
object travels across the circle (you can imagine a sort of weightlessness at the top).

At the top of the circle, the force of gravity acts towards the center of the circle, reducing the
tension required for the necessary centripetal force to its minimum.

At the bottom of the circle, the force of gravity acts against the necessary centripetal force, thus the
tension required at this point is at its maximum.

One assumption of this is that the object travels at a constant speed which is often not the case in
vertical circular motion.

Since a vertical circle involves the object changing in height, it logically follows that the object’s
gravitational potential energy increases as it goes up in the circle and decreases as it moves
down. The law of conservation of energy states that energy cannot be created or destroyed. Hence,
the total energy of the object is shared between gravitational potential energy and kinetic
energy. This is why the object actually changes speed during movement.

Minimum speed:

At the top of the circle, the object has traded most of its kinetic energy for gravitational potential
energy, hence its speed is at a minimum. Since centripetal force is required for an object to move
in a circular path, and tension cannot be negative (i.e. an object is not supported by a string in
compression). This means that the slowest speed the object can travel at is when there is no
tension in the string at the top of the circle, which is when the centripetal force equals the
gravitational force acting on the object:

𝑣𝑚𝑖𝑛 = 𝑔𝑟

Maximum speed:

At the bottom of the circle, the object has traded most of its gravitational potential energy for
kinetic energy, hence its speed is at a maximum. Maximum speed is roughly given by:

𝑣𝑚𝑎𝑥 = 5𝑔𝑟
6.2.1 Newton’s Law of Gravitation

Brilliantly, Newton hypothesized that the force responsible for the falling apple is the same as the
force acting on a planet as it moves around the Sun and that all objects exert an attractive (can not
be negative) force on each other because of their mass, and this force is called the gravitational
force.

Newton’s law of gravitation states that the gravitational force experienced between two
objects is proportional to the mass of the objects and inversely proportional to the square
of the distance between the objects. This is given by:

𝑀1𝑀2
𝐹=𝐺 2
𝑟

where M1 and M2 are the masses of the attracting bodies, r the distance between their centers of
mass and G a constant called Newton’s constant of universal gravitation. It has the value
G = 6.67 × 10−11 N m2 kg−2. The direction of the force is along the line joining the two masses.

The magnitude of the force on each mass is the same. This follows both from the formula as
well as from Newton’s third law.

6.2.2 Gravitational field strength

A mass M is said to create a gravitational field in the space around it. This means that when
another mass is placed at some point near M, it ‘feels’ the gravitational field in the form of a
gravitational force. We define gravitational field strength as follows.

The gravitational field strength at a certain point is the gravitational force per unit mass
experienced by a small point mass m placed at that point.

This is given by:

where F is force and m is mass.

Turning this around, we find that the gravitational force on a point mass m is F = mg. But this is
the expression we previously called the weight of the mass m. So we learn that the gravitational
field strength is the same as the acceleration of free fall.

This can be combined with Newton's law of gravitation to form an equivalent equation for the
gravitational field strength linking the mass of the object around which the gravitational field has
formed, and the distance from the center of mass of that object

In the equations below, M will be the mass of the object around which the electric field has
formed, and m will be the hypothetical small point mass.

𝑀
𝑔=𝐺 2
𝑟

This shows that the gravitational field strength is proportional to the mass of the object, and
inversely proportional to the square of the distance from the center of mass of the object.

As force is a vector, the gravitational field is also a vector, with the units newtons per kilogram (N
kg-1), and as the force is always attractive, the gravitational field is always in the direction of the
mass.

6.2.3 Orbital Motion

The figure above shows a particle of mass m orbiting a larger body of mass M in a circular orbit of
radius r. To maintain a constant orbit there must be no frictional forces, so the only force on
the particle is the force of gravitation. The force of gravitation provides the centripetal force
for this object. Therefore,
 2
𝑚𝑣 𝐺𝑀𝑚
𝑟
= 2
𝑟

Canceling the mass m and a factor of r leads to:

𝐺𝑀
𝑣= 𝑟

2π𝑟
This gives the speed in a circular orbit of radius r. But we know that 𝑣 = 𝑇
. Squaring this and
the previous formula for velocity (to remove the square root) gives:

2
4π𝑟 𝐺𝑀
2 = 𝑟
𝑇
2 3
2 4π 𝑟
⇒𝑇 = 𝐺𝑀

This shows that the period of planets going around the Sun is proportional to the 3/2
power of the orbit radius. Newton knew this from Kepler’s calculations, so he knew that his
choice of distance squared in the law of gravitation was reasonable.