A2 Physics Circular Motion

A2 Physics Circular Motion

1. Describing circular motion

   • When objects move in a circular path instead of a linear one
     • Wheels of a car
     • Hand of a clock
     • Spinning DVD drive
   • Around the clock
     • We can predict the position of the second hand
       • If it moves 360 in one minute, it will move 6 in one second
     • Angular displacement
       • The angle through which the object has moved around a circle
       • Measured in radians

2. Angles in radians

   • Radians are calculated as length of arc/radius
   • Since both measurements are in meters, radians is a ratio that is dimension less
   • 360 degrees = 2 pi
     • Circumference/radius = 2 pi r / r = 2 pi
   • Defining the radian
     • One radian is the angle subtended at the center of a circle by an arc length equal to the radius of the circle
     • To convert degrees to radians: x times pi / 180
     • To convert radians to degrees: x times 180 / pi

3. Steady speed, changing velocity

   • Speed is a scalar quantity whilst velocity is a vector quantity
   • The speed of an object around a circle can remain the same
     • But since the direction of the object is constantly changing, its linear velocity is constantly changing
       • We instead use angular velocity as that remains constant because it moves through the same angle each second

4. Angular velocity

   • The rate of change of angular displacement
   • Greek symbol omega or (w)
   • Measured in radians per second
   • Relating velocity and angular velocity
     • With the example of the second hand clock, each point on the hand has the same angular velocity but different points have different velocities i.e. the tip moves faster
     • Speed = angular velocity x radius

5. Centripetal forces

   • Because its velocity is constantly changing, that means that objects in circular motion must have an acceleration
     • This is the case even though the speed of the object can remain the same
   • Keeping Newtons first law in mind, there is a force since the velocity is not constant
     • If there is a force and the object in circular motion has a mass, there must be an acceleration
       • Experiment: rubber bung. If a rubber bung is swung around in a circle, the tension in the string is the unbalanced force and is what keeps the bung from flying at a tangent. The tension is directed towards you i.e. the center of the circle
         • The centripetal force is the force that is directed towards the center
           • This doesn't mean that it is a force on its own, but an adjective to describe the force. The force could be supplied by things like gravitational acceleration or tension
   • Vector diagrams
     • The change in the velocity (because velocity has direction) can be determined using a vector triangle and shows the difference between initial and final velocity
       • The change in velocity is directed at the center of the circle.
         • Acceleration is change in velocity so the acceleration must also be directed in the circle
   • Acceleration at a steady speed
     • An object can move at a steady speed because it requires a force in the direction of its velocity to speed up or slow down
       • However, we have a force 90* to the direction and so it has no component in the required direction since Fcos90 = 0
         • Also, this can be shown using work done. Work done is the force x displacement
           • Since there is no displacement in the direction of the centripetal force, there is no work done, KE must remain the same so speed must remain the same
   • Understanding circular motion
     • Newton's thought experiment: if we use a large cannon to fire an object horizontally and the object goes too slow, gravity will pull it down and will eventually land.
       • If too fast, the object will keep to its linear velocity and go off into space
         • If we get the speed just right, it will be pulled down towards the Earth but since the Earth as a curved surface it will continually fall towards the planet and that is how an object enters circular orbit

6. Calculating acceleration and force

   • Change in angle = change in velocity/velocity
     • Change in angle x velocity / change in time = change in velocity / change in time
       • a = vw
         • a = w^2r
         • a = v^2/r
   • Newton's second law of motion
     • F = ma
       • a = v^2/r
         • F = mv^2/r
   • Calculating orbital speed
     • F = mg
       • F = mv^2/r
         • mg = mv^2/r
           • g = v^2/r

7. The origins of centripetal forces

   • A car cornering on a level road
     • The road provides two forces. The normal reaction force which balances the weight mg of the car so it has no acceleration in the vertical acceleration and since it is moving to the let there must a centripetal force since the friction between the tyres allows it to follow the circular motion
   • The forces acting on a moving object are not balanced i.e. resultant force. An object moving along a circular path is not in equilibrium and the resultant force is the centripetal force
   • A car cornering a banked road
     • The normal contact force has a horizontal component which can provide the centripetal force. The vertical component of N balances the car weight
   • An aircraft banking a turn
     • The vertical component of the wings life force acts against its weight while the horizontal component provides the centripetal force
   • A stone being whirled in a horizontal circle
     • The vertical component of the tension T acts against the stone weight so the horizontal component provides the centripetal force