Circular motion

Circular motion is where an object goes around in a circle. You can imagine the object movement along the circumference of a circle or rotation along a circular path. Circular motion can be uniform with a contact angular rate of rotation or constant speed or non-uniform where the rate of change of rotation varies. This article explores uniform circular motion. In this article we shall explore quantities like; velocity along the path, acceleration, angular velocity, angular displacement, and the force causing the acceleration. Below is a list of formulae that you should be familiar with;
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The SUVAT equations can also be adapted to be used in constant circular motion;
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…where α is the constant angular acceleration, ω is the angular velocity, ω₀ is the initial angular velocity, θ is the angle turned through (angular displacement), θ₀ is the initial angle and t is the time taken to rotate from the initial state to the final state.
In this article we shall go through the formulae above while explaining the quantities involved.
An object rotating at a steady rate is said to be in uniform circular motion. The motion of the object in a circle is at a constant speed as shown in the animation below;
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For an object in circular motion there is always a constant change in velocity around the circular path however the speed of the object may be constant. This is because the direction of the object is ever changing. Since the motion/velocity of the object is changing there has to be acceleration. An object moving in a circle is always accelerating. An object undergoing uniform circular motion is always with a constant speed even though the object is experiencing acceleration.
An object undergoing uniform circular motion and thus acceleration is under the influence of a net force acting towards the center of the circle. The force is known as a centripetal force. The force is always perpendicular to the velocity.

Velocity

We know that;
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Consider a point on the perimeter of a wheel of radius r rotating at a steady speed;
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The point on the wheel travels a distance equal to the circumference of the wheel. We know that the circumference of the wheel is;
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The time it takes for the point to travel a distance of 2πr is equal to the time of one revolution, the time period (T), so we can conclude that;
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We know frequency is the inverse of time period where f=1/T therefore we can also write that;
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Angular speed and displacement

In this section we shall explore angular speed and displacement. Consider the point on a wheel with diameter 130m such as the London eye.
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On the London eye; each full revolution takes 30 minutes. That must mean that each capsule takes the passengers through an angle of 0.2° each second because;
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This is the same as (π/900 radians per second). This is proof that for any object in uniform circular motion, the object turns through an angle of 2π/T each second where T is the time taken for 1 complete rotation; In other words;
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The angular displacement of the object in time t is therefore given by θ (in radians) 2πt/T = 2πft where T is the time for one rotation and f = 1/T is the frequency of rotation (number of rotations in a second)
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The angular speed, ω, is defined as the angular displacement per second therefore;
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Remember the unit of angular speed ω is given in radian per second (rad s^-1)
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Centripetal acceleration

The velocity of an object in circular motion at constant speed continually changes direction. The object must therefore accelerate. The velocity of the object is always along the tangent to the circle at that point. The change in the direction of the velocity is toward the centre of the circle. So its acceleration is towards the centre of the circle that is why is referred to as “centripetal acceleration”. Centripetal means ‘towards the centre of the circle’
For an object moving at constant speed in a circle of radius r, it can be shown that;
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Notice we can replace v with the speed formula to get;
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You can read about how the centripetal acceleration formula is derived by clicking here.
The centripetal acceleration formulae can be expressed as;
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This is because;
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Centripetal force

There is a wide range of centripetal force applications you may encounter in everyday life. The solar system and many other physical systems are based on centripetal acceleration.
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For any object to maintain circular motion i.e; to move around in a circular path it must be acted on by a resultant force which changes its direction of motion.
The resultant force on an object moving round a circle at constant speed is referred to as the ‘centripetal force’ because it acts in the same direction as the centripetal acceleration, which is towards the centre of the circle.
There are lots of applications where centripetal force may be observed;

• Centripetal force can be seen in an object whirling round on the end of a string where the tension in the string is the centripetal force;
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• For a satellite moving round the Earth, the force of gravity between the satellite and the Earth is the centripetal force.
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• A planet orbiting the sun, the force of gravity between the planet and the sun is the centripetal force
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• A capsule on the London eye, the centripetal force is the resultant of the support force on the capsule and the force of gravity on it.
• A magnetic field can be used to bend a beam of charged particles (e.g electrons) in a circular path. The magnetic force on the moving charged particles is the centripetal force.
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Circular motion is the result of a resultant force which acts towards the centre of the circle. The resultant force is the centripetal force and therefore causes a centripetal acceleration. For any object moving at constant speed v along a circular path of radius r,
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…where ω = v/r. Using Newton’s second law for constant mass we know that; F=ma, therefore;
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