Circular Motion A Review When we see an object carrying out circular motion, we know that there must be force acting on the object, directed towards the center of the circle. When you look at the circular motion of a ball attached to a string, the force is provided by the tension in the string. When the force responsible for the circular motion disappears, e.g. by cutting the string, the motion will become linear.

Going round in circles Speed may be constant But direction is continually changing Therefore velocity is continually changing Hence acceleration takes place

Centripetal Acceleration Change in velocity is towards the centre Therefore the acceleration is towards the centre This is called centripetal acceleration

Centripetal Force Acceleration is caused by Force (F=ma) Force must be in the same direction as acceleration Centripetal Force acts towards the centre of the circle CPforce is provided by some external force – eg friction

Examples of Centripetal Force Friction Tension in string Gravitational pull

Centripetal Force 2 What provides the cpforce in each case ?

Centripetal force 3

Circular Motion Calculations Centripetal acceleration Centripetal force

Period and Frequency The Period (T) of a body travelling in a circle at constant speed is time taken to complete one revolution - measured in seconds Frequency (f) is the number of revolutions per second – measured in Hz T = 1 / f f = 1 / T

Angles in circular motion Radians are units of angle An angle in radians = arc length / radius 1 radian is just over 57º There are 2π = 6.28 radians in a whole circle

Angular speed Angular speed ω is the angle turned through per second ω = θ/t = 2π / T 2π = whole circle angle T = time to complete one revolution T = 2π/ω = 1/f f = ω/2π

Force and Acceleration v = 2π r / T and T = 2π / ω v = r ω a = v² / r = centripetal acceleration a = (r ω)² / r = r ω² is the alternative equation for centripetal acceleration F = m r ω² is centripetal force

Circular Motion under gravity Loop the loop is possible if the track provides part of the cpforce at the top of the loop ( S T ) The rest of the cpforce is provided by the weight of the rider

Weightlessness True lack of weight can only occur at huge distances from any other mass Apparent weightlessness occurs during freefall where all parts of you body are accelerating at the same rate

Weightlessness This rollercoaster produces accelerations up to 4g (40m/s²) These astronauts are in freefall Red Arrows pilots experience up to 9g (90m/s²)

By Farahin Choudhury and Summaiya Rehman Circular Motion

Axis: the straight line around which rotation takes place. Rotation: when an object turns about an internal axis. Revolution: when an objects turns about an external axis. Linear speed: the distance moved per unit time. It is greater on the outer edge of a rotating object than it is closer to the axis. Tangential speed: speed of something moving along a circular path, since the direction of motion is always tangent to the circle. It depends on rotational speed and the distance from the axis of rotation. Rotational speed: the number of rotations per unit of time. The rotational speed is always the same for all the objects, regardless of where they are located. Centripetal Force: any force that causes an object to follow a circular path. Centripetal means "center-seeking" or "toward the center"

Real Life Examples A merry-go-round rotates around its axis. The people sitting in the merry-go-round revolve around the same axis. The person sitting towards the outer edge of the merry-go- round has greater tangential speed than the person sitting towards the middle. Everyone on the merry-go-round has the same rotational speed.

Real life example: the YO-YO As the string unwinds, the yo-yo rolls down one side of the string. It then rolls back up the other side of the string, while simultaneously causing the string to wind up again The potential energy of the yo-yo at the top of its motion is converted to rotational kinetic energy as it falls, and then back to gravitational potential energy again as it rises The velocity at the end of the string is tangential to the orbit, that is, horizontal

Average (Linear) Speed=2  r/T  r=radius and T=period (also called Tangential speed) Centripetal Acceleration=v^2/r  v=velocity and r=radius Net Force=ma  m=mass and a=acceleration (this is Newton’s 2 nd Law) Centripetal Force=mv^2/r  m=mass, v=velocity, and r=radius T=1/f and f=1/T  f=frequency and T=period Rotation Speed=  rotations/  time

A short Demonstration… kn2tB-9Ehttp:// kn2tB-9E

Sample Word Problem A small patch on a tire is located 20 centimeters from its axis of rotation. If it makes 1 revolution in 12 seconds, what are its tangential and rotational speeds? (Hint: USE THE EQUATIONS!!!!!)

Axis: the straight line around which rotation takes place Rotation: when an object rotates around an internal axis Revolution: when an object turns around an external axis

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Linear Speed: is the distance moved per unit of time (points on the outside of the circle travel further than points near the axis) Rotational Speed: is the # of rotations per unit of time (all parts on a circle rotate in the same amount of time); “RPM” »Linear Speed = radial distance x rotational speed

On a merry-go-round the horses on the outer rail are located 4 times as far from the horses on the inside rail. If a girl sits on a horse on the inside rail at a rotational speed of 3 RPM and a linear speed of 4 m/s, what will be the rotational speed and linear speed of her brother who is on a horse on the outer rail? L speed = rad distance x rot speed = 4 x 3 = 12 m/s His rotational speed is the same at 3 RPM. His linear speed however is a quicker 12 m/s.

Centripetal Force is any force that causes an object to follow a circular path. Gravitational and electrical forces act across empty space as centripetal forces. Ex: A gravitational force directed towards the center of the earth holds the moon in circular orbit around the earth. Ex: Electrons revolving around the nucleus of an atom are held in orbit by an electrical force.

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