How Motion in a Circle Affects System Dynamics 

In engineering applications, circular motion is encountered in various systems such as rotating machinery, turbines, gears, flywheels, and vehicle dynamics. Unlike linear motion, circular motion involves continuously changing direction, resulting in dynamic effects that must be analysed to ensure safe and efficient design.

Understanding these dynamics is essential for designing mechanical systems subjected to rotational or curved motion, where unbalanced forces or vibrations can lead to mechanical failure or reduced performance.

Examples of circular motion include:

•  an artificial satellite orbiting the Earth at a constant height, 
• a stone which is tied to a rope and is being swung in circles, 
• a car turning through a curve in a racetrack, 
• a gear turning inside a mechanism 
• a motor vehicle wheel.

Consider a body moving in a circular path with a constant angular velocity, as follows:

The angular velocity, ω, is defined as the rate of change of angle,θ , with respect to time, t, and is analogous to its linear velocity counterpart:

Linear Velocity v (m/s) = ^∆x⁄_∆t

Angular Velocity ω (rad/s) = ^∆θ⁄_∆t

Constant Angular Velocity

At any given instant in time, t, there will be a tangential velocity, v_t, which is perpendicular (at right angles to) the radius, r of the circle, which is defined as:

v_t = ω r

The above equation is the ‘link’ between the linear and the angular domain.

Even though we are considering constant angular velocity, there will be an acceleration towards the centre of the circle, a_c. This acceleration is known as the centripetal acceleration.

Consider a vehicle travelling around a bend which forms part of a circle. As you are probably aware, you, as a passenger of such a vehicle, will experience a tendency to feel like you are being forced out towards the outside of the circle. This is the reaction force associated with the centripetal acceleration and is known as the centrifugal force due to the centrifugal acceleration. The centripetal acceleration (and its associated centripetal force) is always towards the centre of the circle and it is the job of the tyres on the vehicle and the chassis stiffness to produce this force as this is the force that holds the vehicle in a circular path.

The centripetal acceleration is given by:

a_c = ^v_t2⁄_r

By Newton’s second law, the associated centripetal force is

F_c = ma_c

F_c = ^mv_t2⁄_r

Example 1

A vehicle of mass 1250 kg travels round a bend of radius 75 m, at 60 km/h. Determine the centripetal force acting on the vehicle.

First, convert the velocity to SI units:

60km/hr = 60,000 m /hr = 60,000 / 3600 m/s = 16.7 m/s

F_c = ^mv2⁄_r

F_c = ^{(1250)(16.7)2}⁄₇₅

F_c = 4630N

Changing Angular Velocity

In this case, the angular velocity is changing, and hence we will also have an angular acceleration, α.

Linear Velocity v (m/s) = ^∆x⁄_∆t

Angular Velocity ω (rad/s) = ^∆θ⁄_∆t

Linear Acceleration, a_t (m/s²) = ^Δv⁄_∆t

Angular Acceleration, α (rad/s²) = ^∆ω⁄_∆t

There will also be a tangential acceleration, a_t, given by:

a_t = α r

Example 2

A 5kg wheel with a radius of 300mm accelerates from rest to 500 RPM in 10 seconds. At time, t = 10s, determine

a) the angular velocity,

b) the angular acceleration 

c) the tangential velocity on the rim of the wheel

d) the centripetal acceleration and centripetal force 

e) the tangential acceleration

Solutions

a) angular velocity

ω = 500 RPM

ω = ⁵⁰⁰⁄₆₀ rev per sec = ⁵⁰⁰⁄₆₀ (2π) rad/s

ω = 52.4 rad/s

b) angular acceleration

α = ^Δω⁄_Δt = ^(52.4-0)⁄_(10-0) = 5.24 rad/s²

c) tangential velocity

v_t = ω r = (52.4)(0.3) = 15.7 m/s

d) centripetal acceleration and force

a_c = ^v_t2⁄_r = ^(15.7)2⁄_0.3 = 822.5 m/s²

F_c = ma_c = (5)(822.5) = 4112.3N = 4.1kN

e) tangential acceleration

a_t = α r

a_t = (5.24) (0.3) = 1.6 rad/s²

It should also be noted that all of the linear equations of motion for constant acceleration (SUVAT) have their angular motion equivalents as summarised below:

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