One radian is the angle subtended at the centre of a circle by an arc of length equal to the circle's radius.
Any angle measured in radians is given by dividing the curved distance along the arc of a circle by the radius of the circle: (theta)=c/r
One revolution=circumference of circle/ radius of circle -> 2πr/r =2πradians
Reasons for using radians: The unit itself is distance divided by a distance and so doesn't have dimensions.
The main reason for using the radian is to do with rotation. A measurment of the rate of rotation of an engine is sometimes quoted in revolution per minute, but scientists measure the rate of rotation, or angular velocity, in radians per second.
The period t of an object in circular motion is the time taken to complete one revolution. It is related to the speed v and the radius r by the equation v=2(pi)r/T
speed=distance/time = circumference of circle/period
A car going round a corner at a constant speed is changing velocity because the direction of the velocity is changing, although its magnitude remains constant.
Acceleration is the rate of change of velocity, so for an object describing a circular path, it means that the acceleration also must be towards the centre of the circular path.
The centripetal acceleration a of an object travelling in a circle of radius r with constant speed v is given by a=v^2/r in a direcion towards the centre of the circle.
The Force required for Centripetal Acceleration
When an object is travelling round in a circle, it is subject to Newton's laws. Newton's laws apply to all objects unless they are travelling near to the speed of light.
The Earth exerts a gravitational force on the Moon and the Moon exerts a gravitational force on the Earth that is equal but opposite (Newton's 3rd law). If Newton's 2nd law is applied to the Moon, we get F=ma where F is the force exerted on the Moon by the Earth, m is the mass of the Moon and a is the acceleration of the Moon in its orbit around the Earth (which is v^2/r). Therefore when Newton's 2nd law is applied to the Moon, we get
The net force causing the centripetal acceleration is often called the centripetal force. Here, the centripetal force is the gravitational pull of the Earth on the Moon.
Examples of Circular Motion - 1
1. Ball on a string travelling on a friction-free surface. The surface is supporting the ball and the support force it supplies is equal and opposite to the weight of the ball. The force that causes the ball to have acceleration v^2/r towards the central peg is provided by the tension T in the string. Therefore, T=mv^2/r
Examples of Circular Motion - 2
2. A conical pendulum: similar to the previous example however there is now no friction-free surface. The ball rotates in a horizontal circle of radius R and the resultant (net) force on the ball must be directed towards the central point. The force the string exerts on the ball needs to do 2 things: its vertical component must be equal and opposite to the weight of the ball, and its horizontal component must accelerate towards P. Tcos(theta)=mg and Tsin(theta)=mv^2/r therefore, tan(theta)=v^2/rg
Field: the region in which a force operates.
The Earth exerts a gravitational force on the Moon. Therefore, the Moon is in the Earth's gravitational field. There is a force pulling the Moon towards the Earth, so the direction of the Earth's gravitational field is directly towards the Earth.
Gravitational field strength at any point is the force acting per unit mass at that point. (N/kg)
Newton's Law of Gravitation - 1
Newton's law of gravitation states: The gravitational force of attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. F (proportional to) Mm/r^2
G= 6.673x10^-11 m^3/kg.s^2
*Remember: The Earth's orbit is not quite circular. It has been assumed that the Sun is stationary however the movement of the Sun due to the Earth going round it is very small because of the Sun's very large mass compared with that of the Earth. A year is not 365 days exactly.
Newton's Law of Gravitation - 2
If F=ma and F=-GMm/r^2, then GMm/r^2=mg
Cancelling m on both sides of the equation gives: g=GM/r^2
how an object becomes a satellite
Keplar's 3rd law: the period squared is proportional to the mean radius cubed. T^3 (proportional to) r^3
F=mv^2/r where v=2πr/T
combining these equations gives:
F=m4π^2r^2/T^2r = GMm/r^2
Finally giving T^2=(4π^2/GM)r^3
since 4π^2/GM is constant, T^2 is proportional to r^3
Geostationary satellites: has its orbit centred on the centre of the Earth, travels from west to east, orbits over the equator and has a period of 24 hours (86400s).
Keplar's law applies to geostationary satellites. T has to be 24 hours, or 86400s, therefore the radius r of the orbit can be worked out. This works out to be 42300km from the centre of the Earth. The height above the Earth will be => 42300km-6400km=35900km
What are they used for? Geostationary satellites are used for telecommunications. They transmitt at high power because they are far away from the Earth.
Low level satellites: not geostationary, usually orbit at about 500km. Because they are always moving, many of them are necessary for complete coverage of the Earth at all times. However, they are cheaper to put into orbit, the Earth can be seen in greater detail in photos, a higher intensity (power per unit area) can be achieved on the Earth's surface. What are they used for? For these reasons, low-level satellites are used as weather satellites, spy satellites, mapping and GPS.
Introduction to Simple Harmonic Motion
S.H.M. refers to motion involving a body that oscillates. Oscillations that take place in S.H.M. are smooth with no variation in amplitude. In S.H.M., the period is independant of the amplitude.
Oscillations that take only a short time: banging a drum, knocking on a door, hitting a nail with a hammer.
Oscillations taking place too fast for us to sense: light and warming effect of Sun.
Oscillations we cannot sense: radio waves, X-rays, microwaes from a mobile phone
Simple oscillation: the swinging of a clock pendulum. If the pendulum is not connected to a driving mechanism then the fricccction will gradually reduce the extent of the swing until it stops. A free oscillation has no driving mechanism and no friction. Once started it would oscillate forever. (practically no oscillation is ever completely free)
Terms used to describe Oscillations
Displacement: x (in metre units (m)), the distance an object moves form its mean (or rest) position: it may be positive or negative.
Amplitude: x0 (in metre units (m)) is the maximum displacement and will always be positive.
Frequency: f (in Hertz (Hz)) is the number of oscillations per unit time at any point
The Period T (in seconds (s)) is the time for one complete pattern of oscillation to take place at any point.
There is a connection between oscillations and circular motion. One oscillation corresponds to one rotation. Using angular measure, this is an angle of 2(pi)radians. If there are f oscillations per unit time, there will be a corresponding angle of 2(pi)f radians per unit time.
Phase difference: Two points that have exactly the same pattern of oscillaion are in phase. Phase difference is the angle in radian between 2 oscillations.
Definition of Simple Harmonic Motion
When the mass is in equilibrium the tension in the spring T equals the weight W of the mass.
When the mass oscillates and is stretched by a distance x, so the upward force the spring exerts on the mass is T + kx. The net force on the mass is therefore: (T + kx) - W = kx upwards, since T=W.
When the mass oscillates and is squashed so the displacement is shorter by a distance y, the tension in the spring is therefore T - ky, and the net force on the mass upwards is: (T - ky) - W = -ky upwards, which is ky downwards.
This shows that the net force on the mass is directly proportional to the displacement but in the opposite direction to the displacement. When the mass goes up the resultant force is downwards and vice versa. This pattern of applied force, and hence acceleration is S.H.M. a = -cx where c is a constant.
x=Asin(2πft) and x=Acos(2πft) where t is the time and A is constant
Graphs of Simple Harmonic Motion
Graphs of Simple Harmonic Motion
therefore, x=Acos(2πft) does satisfy the s.h.m. equation.
Energy in Simple Harmonic Motion
Damping: deliberately reducing the amplitude of an oscillation.
The damping forces are usually the forces of friction and air resistance. If only small damping forces exist - light damping - the period of the oscillation is almost unchanged but the amplitude gradually decreases. As the damping forces increase, the amplitude decreases considerably and the period increases slightly.
Eventually for heavy damping, no oscillation occurs, and the body slowly moves back to its equilibrium position. The cross over situation between oscillation and no oscillation is called critical damping.
Examples of damping: The suspension system of a car: the springs on each wheel of the car gives the car a smoother ride but without damping, the bounce of the springs after each bump would be unpleasant. In the design stages of a concert hall, models are tested to see how long it takes for sound levels to drop: if it is too short the hall seems dead, if it is too long the sound gets fuzzy. Damping is used to ensure that the oscillations do not continue for past the appropriate time.
Resonance - 1
Resonance: the build up of a large amplitude oscillation when the frequencies of vibrating objects match.
Example: a singer who shatters a wine glass by singing at just the right frequency to match a natural frequency of the wine glass/ Tacoma Narrows bridge which collapsed due to winds causing vibrations that built up over several hours.
In these cases there are at least two vibrating objects. There is the object causing the effect: its frequency is called the driver frequency. There is also the driven obeject which has a natural frequency of vibration. A large-amplitude is built up when the two frequencies are the same: resonance.
Resonance - 2
Effect of damping on resonance:
Practical uses of resonance: When you tune in a radio or TV programme, you are adjusting the resonant frequency of the radio or TV receiver to the frequency of the transmitted signal.