Circular Motion

PHYSICS SYLLABUS

APPLIED MATHS SYLLABUS

When we talk about circles we usually need to be able to talk about angles. In maths we are use to talking about degrees, however in physics and applied maths we often prefer to deal in a different unit of 'Angular Separation' rads. Just like the difference between other units there is a rate of conversion,

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So one complete rotation, as you already know is equal to 360o but now you know it is equal to 2 x 3.14 rads. One rotation is equal to 6.28 rads, or in other terms 1rad = 57.2727o, while that converts it into values you know, it might not actually be the most convenient way. Many of the questions are prepared as to make the answer quite convenient to leave in terms of pi.

so in terms of rads we see that

where s is the length of the arc the particle passes through,

so by rearranging we could get

which is an expression for the distance

We have previously described that acceleration is the rate of change of velocity.

And that velocity is a vector quantity of speed with direction.

So if something is accelerating it could be due to 1 of 2 factors

  1. A change in its speed
  2. A change in its direction.

If something is rotating / going around in a circle it must be changing its direction, thus the body would be accelerating (in the true meaning of the word)

If something is accelerating it must be caused by a force (Newtons 2nd Law)

The force which causes this movement must be toward the center, this is called centripetal force. But Centripetal force is just another force that causes motion to go towards the center, friction in a car going around a bend, tension in a rope, gravity of a satellite, or resultant.

We might be on one of those fair rides that spins round, we get pushed into the wall. Sometimes on these rides the floor disappears and yet we are stuck to the wall, this is called centrifugal force. This is not a real force !

This force appears due to the 1st of Newtons laws of motion, that an object continues in a straight line at a constant velocity, unless acted on by an external force, in this case centripetal force, the force that makes the ride spin around.

This is called inertia, that a body wants to remain in the state it is in. Think about driving in the car, imagine a ball in the back seat, what happens the ball when you go left? or brake suddenly.

So were off the mark with this Carousel and

  1. make sure to vary the period (the time it takes to do 1 complete revolution),
  2. the length of the strings
  3. and the mass of the particle

Centripetal force required to maintain uniform motion in a circle.

Centripetal force is caused by whatever causes the object to move in a circle, sometime this will be the tension in a rope .. etc..., some times this will be the reaction force upon the particle from the surface upon which it travels. If the body travels on the inside of a circular surface.

NASA do a video on it

look in the for educators section

check for centripetal acceleration

http://www.nasa.gov/audience/foreducators/topnav/materials/listbytype/Angular_Momentum_Inertia.html

therefore by dividing both sides by the mass, m, we get

we get

8. Angular velocity. Definition of angular velocity ω.

Angular velocity ω is the angle through which an object travels through per second

Angular velocity is the rate at which the body revolves around its center. For a moment lets think about a C.D. spinning in a CD player. Does each part of the CD cover the same distance ? Which part of the CD covers more distance? However the any line from the center to a point on the outside of the CD completes 1 revolution in the same amount of time.

The angle swept out by the body over the time taken to complete the angle.

as we saw before

Derivation of v = rω

where s is displacement, displacement divided by time is velocity so if we divide both sides by time we get

Use of a = rω 2, F = m rω 2

now putting this into the formula earlier for acceleration gives us

Period is T, this is the time taken to do a complete revolution. 1 complete revolution is 360o or in radians 2

Uniform motion in a circle without gravitational forces.

Conical pendulum.

The conical pendulum question is very popular type of question in the exams. This is where a pendulum hangs from a point but the bob is held up by a surface and so makes an angle between the point of suspension and the mass.

Questions of this type are usually solved by breaking the Tension down into its components ║ & ┴ to the surface, and remember that the forces up = forces down, usually Tcos + R = W

Circular orbits.

Demonstration of circular motion.

"Moment of inertia," or MOI, is a property of physics that indicates the relative difference in how easy or difficult it will be to set any object in motion about a defined axis of rotation. The higher the MOI of an object, the more force will have to be applied to set that object in a rotational motion. Conversely, the lower the MOI, the less force needed to make the object rotate about an axis.

golf.about.com

A very good demo video here!

http://hyperphysics.phy-astr.gsu.edu/HBASE/mechanics/rstoo2.html#c1

Keplers laws

http://www.walter-fendt.de/ph14e/keplerlaw1.htm

  1. The orbit of every planet is an ellipse with the Sun at one of the two foci.
  2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Satalites in action

http://www.imdo.ie/followthefleet/

Force Diagrams

When drawing out the suitable forces acting on the particle, it is important to note that the moving particle is actually causing the centripedal force and thus cannot be said to be acting on the particle.

Limiting forces

When a particle is held by a string or by friction the force holding it often has a limit this limit is overcome after

centripedal force = limiting force, and so max speed or tension is at a maximium value at this equation.

If a particle is on the outside of another circle the particle will 'fall' off when R = 0

A satellite rotates about a planet, the period T (the time take to complete one orbit) of such relies on the distance that the satellite must cover and the speed at which it is travelling

There are different types of questions

Strings

Where a string holds the particle there is a Tension in the string. The Tension has 2 components, 1 parallel to the circle being cast this component of the tension is = the centripetal force. If this object rotates on a surface then there is also a Reaction occurring too. The force R = mg - T calculate this perpendicular force to solve the problem for w

Bowls ...

There is no Tension and the Reaction force is at right angles to the surface of the bowl at that point. Or if you prefer at a tangent. so the R component paraalel is the centripedal force & the R┴ = the weight ... mg

Vertical Circles,

remember that the weight and the force holding the object in a circular path will work opposite and with each other at the top and bottom, respectively, of the circles.

remember that as things that fall, they change their energy from potential energy to kinetic energy ....

If a particle is on the outside of another circle the particle will 'fall' off when R = 0

Appropriate calculations.

The average length of the Moons Period is 29 days, 12 hours, 44 minutes and 2.9 seconds.

To Do Applied Maths Questions