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Circular Motion

Centripetal Acceleration

Car in circular motion
When a car goes around a curve, it is accelerating towards the centre of a circle, and the passengers feel a force pushing them towards the outside of the vehicle.

A mass moving in a circle of radius R, and period of revolution T has a speed v:

$$v = {2πR}/T$$

The angular speed of the object is the angle swept in a time interval. Since one revolution is $2π$ radians, one revolution is made in $T$ seconds, where T is the period. The angular speed $ω$ is:

$$ω = {2π}/T$$

Even though the speed is constant, the velocity (vector) is constantly changing, and its vector is pointing along the tangent to the circle, or perpendicular to the radius at that point. This changing velocity means there is constant acceleration directed towards the centre of the circle.

$$a_c = {v^2}/R$$

where $a_c$ is the centripetal acceleration, v is the tangential velocity, and R is the radius of the circle.

Centripetal Force

Carousel joyride
The centrifugal effect is the sensation of a force, but is really a result of inertia and the centripetal force.

Angular Momentum

A lever enables people to lift heavy objects. A lever works by a long handle over a fulcrum close to the mass to be moved.

Archimedes, the famous Greek scientist who lived in the 3rd century BCE, is reputed to have said of levers: 'Give me a place to stand, and I will move the world'. Of course, without a fixed fulcrum, no lever would move anything.

A seesaw demonstrates the moments involved in levers
A lever allows a smaller mass to move a larger, if it is sufficiently far from the fulcrum.

He had discovered that if he wanted to move a heavy weight, placing a piece of wood under it, and creating a pivot point with a rock, he could make a lever, which made lifting a lot easier.

A lever makes use of the fact that the force applied increases the further it is from the pivot point, or fulcrum, causing a rotation movement. This force of moment is called torque.

Tools like hammers and spanners use force of moment, or torque, to apply greater force than would be otherwise possible.

A hammer is also a lever

The hammer can lever up a nail because the force applied to the handle is much further than the nail from the fulcrum point on the head.

A spanner is also an example of a tool which utilises the lever principle to good effect. Trying to turn a bolt with the fingers is impossible. But when we put a spanner on it, we find our arm has enough strength to cause the bolt to rotate. The longer the spanner the more force we can apply.

Another example of a lever is the seesaw. An adult can find equilibrium with a child by sitting closer to the central fulcrum.

The Maths

The moment of force, or torque, which is applied by a mass m1, at distance d1, is F.d1 = m1gd, where d is the distance from the fulcrum.

The mass m2 at distance d2that can be moved must be applying a torque to the lever in the opposite direction less than the torque of mass m1.

Content © Andrew Bone. All rights reserved. Created : September 9, 2013 Last updated :February 27, 2016

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