When an object travels in a complete circle it moves through an angle of 360º, although the distance it travels depends on the radius of the circle. An alternative unit to the degree for measuring angular displacement is the radian, defined as the arc length divided by the radius of the circle.
For a complete circle (360º) the arc-length is just the circumference of the circle (2∏r). Divided by the radius r fives 2∏, which is the amount of radian in a complete circle.
Speed in a Circle 1:
When an object is moving in a circular path its velocity is always at right angles to a radial line, a line joining it to the centre of the circle. Two objects such as two points on a CD, at different distances from the centre each complete the same number of revolutions in any specified time but they have different speeds. They have the same angular velocity.
Angular velocity, ω,is defined as:
The rate of angular displacement with time.
ω = Ø / t
The relationship between angular velocity and linear velocity for an object moving in a circle of a radius r is:
V = rω
Speed in a circle 2:
The time period for one resolution, T, is equal to the angular displacement (2∏ radians) divided by the angular velocity.
T = 2∏/ω
The frequency of a circular motion, f, is equal to 1/T
F = 1/T = ω/2∏
Any object that changes its velocity is accelerating. Since velocity involves speed and direction, a change in either of these quantities is acceleration. An object that is moving at a constant speed in a circle is therefore accelerating as it is continually changing in direction. The change in direction is always towards the centre of the circle.
The acceleration of an object moving in a circle:
· Is called centripetal acceleration
· It is directed to the centre of the circle
Centripetal acceleration, a, is the acceleration of an object moving in a circular path:
a = v²/r
Where v, is the linear velocity, and ω is the angular velocity and r is the radius of the circle.
Theme Park Rides:
Some theme park ride involve motion in a vertical circle. Part of the thrill of being on one of these rides is the apparent change in weight as you travel round in a circle. This is because the force that you feel as you sit or stand is not your weight, but the normal contact force pushing up. If the forces on you are balanced, the force is equal in size to your weight, but it can become bigger or smaller if the forces are unbalanced, causing you to feel heavier or lighter.
At the bottom of the circle:
· The person feels heavier than usual as the size of the normal reaction force is equal to the person’s weight plus the centripetal force required to maintain circular motion. mv²/ r + mg.
At the top of the circle:
· The person will feel lighter than usual as the person’s weight is providing part of the centripetal force required to maintain circular motion. The rest comes from the normal reaction force. mv²/ r – mg.
· If the speed of the ride is such that the required centripetal force is the same as the person’s weight then he or she feels “weightless” as the normal reaction force is zero.
A field is a region of space where forces are exercted on objects with certain properties. There are three main forces:
1. Gravitational fields affect anything that has mass.
2. Electrical fields affect anything that has charge.
3. Magnetic fields affect permanent magnets and electric current.
· Gravitational forces are always attractive – the Earth cannot repel an object.
· The Earth’s gravitational field pull acts towards the centre of the earth.
· The Earth’s gravitational field is radial, the field lines become less concentrated the future the distance from Earth.
The force exerted on an object in a gravitational field depends on its position. The less concentrated the lines, the smaller the force If the gravitational field strength at any point is known, then the size of the force can be calculated:
The gravitational field strength (g) at any point in a gravitational field is the force per unit mass at that point:
g = F/m
- F is the force experienced by a mass, m when it's placed in the gravitational field. Divide F by m and you get the force per mass.
- g is a vector quantity, always pointing towards the centre of the mass whose field you are describing (because gravitational force acts in that direction. This means g is negative.
- g is just the acceleration of a mass in a gravitaional fiels. It oftern called the acceration due to gravity.
In a radial field, g is inversely proportional to r².
Point like masses have radial gravitational fields, the strength of the gravitational field, g, decreases the further away you are from the centre mass.
g = -GM/ r²
In studying gravitation, Newton concluded that the gravitation attraction force that exist between any two masses:
1. Is proportional to each of the masses.
2. Is inversely proportional to the square of their distances apart.
Newton’s law of gravitation describes the gravitational force between two point masses. It can also be written as:
Where G is the universal gravitational constant and has the value of 6.7x10^-11 Nm²Kg-²
G and g 1:
Newton’s law of gravitation can be used to work out the value of the force between two objects. It can also be used to calculate the strength of a gravitational fiels due to a spherical mass such as the Earth or the Sun.
A small object, mass m, placed with in the gravitational field on Earth, mass M, experiences a force, F, given by:
Where r is the separation of the centre of the mass of the object and the Earth.
G and g 2:
It follows the definition of gravitational field strength as the force per unit mass that the field strength at that poit, g, is related to the mass of the Earth by the expression:
g= F/m =GM/ r²
Gravitational field strength is a property of any point in the field. It can be given a value if or not a mass is placed at that point. Like gravitational force, beyond the surface of the Earth the value of g follows a inverse square law. Because an inverse square law applies to values g when the distance is measured from the centre of the Earth, there is little change in the value close to the Earths surface. Even when flying an aircraft there is no noticeable change in the value g.
Potential and Potential Energy 1:
When an object changes its position relative to Earth, there is a change in potential energy given by ΔEp = mgΔh. It is not possible to place an absolute value on the potential energy of an object when h is measured relative to the surface of the Earth. To similar objects placed at the top and the bottom of a hill have different potential energies, but relative to the ground the potential energy is zero for both objects.
Absolute values of potential energy are measures relative to infinity. In this context, infinity means ‘at a distance from Earth where gravitational field strength is so small as to be negligible’.
Using this reference point:
1. All objects at infinity have the same amount of potential energy, zero.
2. Any object closer than infinity has a negative amount of potential energy, since it would need to acquire energy in order to reach infinity and have zero energy.
Potential and Potential Energy 2:
Just as gravitational field strength is used to place a value n the gravitational force that would be experienced by a mass at any point in a gravitational field, the concept of gravitational potential is used to give a value for the potential energy.
The gravitational potential energy at a point in a gravitational field is the potential energy per unit mass placed at that point, measure relative to infinity.
When an object is within the gravitational field of a planet, it has a negative amount of potential energy measured relative to infinity. The amount of potential energy depends on:
1. The mass of the object.
2. The mass of the planet.
3. The distance between the centres of mass of the object and the planet.
The gravitational potential energy measured relative to infinity of a mass, m , placed within the gravitational field of a spherical mass M can be calculated using:
Ep = - GMm / r
Where r is the distance between the centres of mass and G the universal gravitational constant. Measured in Joules. Since gravitational potential is the gravitational energy per unit mass placed at a point in a field, it follows that:
Gravitational potential, V, is given by the relationship: Measured in J Kg -1.
V = Ep / m = GM/r