circular motion.docx

Chapter 8 Circular Motion

Angle (in radians)

Angular Velocity =

Angular Velocity is the rate of change of angle with respect to time. Angular Velocity is measured in radians per second, (rad/s).

If an object is moving in a circle at constant speed, it is accelerating.

This is because while its speed is not changing, its velocity is. Velocity is defined as speed in a given direction, so if direction is changing, even though speed is not, then the velocity is changing, therefore the object is accelerating.

θ=s

r

¿

❑

Centripetal Force

The force - acting in towards the centre - required to keep an object moving in a circle is called Centripetal Force.

It depends on three factors; greater force is required if –  The object is moving with greater velocity  The object has a smaller radius to turn  The object has a higher mass

Centripetal Acceleration

If a body is moving in a circle the acceleration it has towards the centre is called Centripetal Acceleration.

Formulae for Centripetal Acceleration and Centripetal Force

but because v = r we also have

And because F = ma we get and also

a=v

2

r

¿¿

a = r



F

=

mv

2

Circular Satellite Orbits

Relationship between Periodic Time and Radius for a Satellite in Orbit

Kepler’s laws of planetary motion were based on observations of elliptical orbits – circular orbits. Kepler’s third law has the following concepts

 All bodies in the solar system attract one another.

 The force between the bodies is proportional to their masses and inversely proportional to the square of the distance between them.

T = periodic time (s)

R= radius of body

T

4

π

Derivation of formula:

Geostationary Satellites

These are satellites which remain (are stationery) over one position of the globe, and their orbit is called a Geostationary orbit.

Exam Questions

G = 6.7 × 10-11 _{N m}2 _kg-2_{, Radius of Earth = 6.4 × 10}6 _m,

Mass of Earth = 6 × 1024 _{kg. Mass of the sun = 2 × 10}30 _{kg. Mass of Saturn = 5.7 × 10}26 kg.

Mean distance of Saturn from the Sun = 7.8 × 1011 _m.

1. [2006][ 2005]

Define angular velocity.

2. [2006]

5. [2005]

Define centripetal force.

6. [2004]

Give an expression for centripetal force.

7. [2004]

(i) Centripetal force is required to keep the earth moving around the sun. What provides this centripetal force?

(ii) In what direction does this centripetal force act?

8. [2009]

A skateboarder of mass 70 kg maintains a speed of 10.5 m s–1 _{until he enters a circular ramp of radius 10 m.}

What is the initial centripetal force acting on him?

9. [2004]

The earth has a speed of 3.0 × 104 _{m s}–1 _{as it orbits the sun.}

The distance between the earth and the sun is 1.5 × 1011 _{m. Calculate the mass of the sun.}

10. [2009]

The moon orbits the earth. What is the relationship between the period of the moon and the radius of its orbit?

11. [2008][2005]

13. [2008]

The international space station (ISS) moves in a circular orbit around the equator at a height of 400 km. (i) Calculate the period of an orbit of the ISS.

(ii) After an orbit, the ISS will be above a different point on the earth’s surface. Explain why. (iii) How many times does an astronaut on the ISS see the sun rise in a 24 hour period?

2014

(i) The Martian moon Phobos travels in a circular orbit of radius 9.4 × 10

_{m around Mars with a period of}

7.6 hours.

Calculate the mass of Mars.

2013 Question 6

(i) State Newton’s law of universal gravitation.

(ii) Explain what is meant by angular velocity.

(iii)

Derive an equation for the angular velocity of an object in terms of its linear velocity when the object

moves in a circle.

The International Space Station (ISS), shown in the

photograph, functions as a research laboratory and a

location for testing of equipment required for trips to the

moon and to Mars. The ISS orbits the earth at an altitude of

4.13 × 10

_{m every 92 minutes 50 seconds.}

(iv)Calculate (

a

) the angular velocity, (

b

) the linear velocity, of the ISS.

(mass of ISS = 4.5 × 10

_{kg; radius of the earth = 6.37 × 10}

_m)

2012 Question 12 (a)

An Olympic hammer thrower swings a mass of 7.26 kg at the end of a light inextensible wire in a circular

motion. In the final complete swing, the hammer moves at a constant speed and takes 0.8 s to complete a

circle of radius 2.0 m.

(i) What is the angular velocity of the hammer during its final swing?

(ii) Even though the hammer moves at a constant speed, it accelerates. Explain.

(iii)

Calculate the acceleration of the hammer during its final swing