Rotational motion
1. Rotational motion:
- ⇒ Angular displacement (θ):
- Angular displacement is the angle through which an object rotates, measured in radians (rad).
- Symbol: θ (theta)
- Units: radians (rad)
- Definition:
- “The angle between the initial and final positions of a rotating object”.
- [math]θ = \frac{s}{r} [/math]
- Where:
- – θ = angular displacement
- – s = arc length (distance traveled by the object)
- – r = radius of the circle (or rotation)
- Angular displacement is used to describe:
- – Rotation of objects, like wheels, gears, and circular motion.
- – Position and orientation of objects in space.
- – Rotational kinematics and dynamics.
- – Examples:
- (1)
- An object moves 4 meters along a circular path of radius 2 meters. Find the angular displacement in radians.Given data:
- Distance traveled by the object (arc length) = s =4m
- Radius of the circle = r = 2m
- Find data:
- Angular displacement = θ =?
- Formula:
- [math]θ = \frac{s}{r} [/math]
- Solution:
-
[math]θ = \frac{s}{r} \\ θ = \frac{4}{2} \\ θ =2 radians[/math]
- (2)
- A car travels 120 meters along a circular track of radius 30 meters. Find the angular displacement in radians.
- Given data:
- Distance traveled by the car (arc length) = s =120m
- Radius of the circular track = r = 30m
- Find data:
- Angular displacement = θ =?
- Formula:
-
[math]θ = \frac{s}{r} [/math]
- Solution:
- [math]θ = \frac{s}{r} \\ θ = \frac{120}{30} \\ θ = 4 radians[/math]
- ⇒ Angular speed (ω):
- Angular speed is the rate of change of angular displacement, measured in radians per second (rad/s).
- Symbol: ω (omega)
- Units: radians per second (rad/s)
- Definition:
- “The rate at which an object rotates, measured as the change in angular displacement per unit time”.
- [math] ω = \frac{∆θ}{∆t} [/math]
- Where:
- – ω = angular speed
- – ∆θ = change in angular displacement
- – ∆t = change in time
- – Examples:
- (1)
- Find the angular speed of a wheel that rotates 60° in 4 seconds.
- Given data:
- Angular displacement = ∆θ = 60°
- Time = ∆t = 4s
- Find data:
- Angular speed = ω =?
- Formula:
- [math] ω = \frac{∆θ}{∆t} [/math]
- Solution:
- [math] ω = \frac{∆θ}{∆t} \\ ω = \frac{60}{40} \\ ω = 15 degree/s [/math]
-
⇒Angular velocity (ω):
- Angular velocity is a vector quantity that describes the rotation of an object. It has both magnitude (amount of rotation) and direction (axis of rotation).
- Symbol: ω (omega)
- Units:
- – radians per second (rad/s)
- Definition:
- “The rate of change of angular displacement with respect to time”.
- [math] ω = \frac{∆θ}{∆t} [/math]
- Where:
- – ω = angular velocity
- – ∆θ = change in angular displacement
- – ∆t = change in time
- Angular velocity is used to describe:
- – Rotation of wheels, gears, and other circular motion.
- – Precession and nutation of rotating objects (like gyroscopes).
- – Rotational kinematics and dynamics.
- – Torque and rotational energy.
- Angular acceleration (α):
- Angular acceleration is the rate of change of angular velocity, measured in radians per second squared ([math] rad/s^2[/math]).
- Symbol: α (alpha)
- Units: radians per second squared ([math] rad/s^2 [/math])
- Definition:
- “The rate at which the angular velocity of an object changes, measured as the change in angular velocity per unit time”.
- [math] α = \frac{∆ω}{∆t} [/math]
- Where:
- – α = angular acceleration
- – ∆ω = change in angular velocity
- – ∆t = change in time
- Angular acceleration is used to describe:
- – Changing rotation speed of wheels, gears, and other circular motion.
- – Acceleration of rotating objects, like engines, motors, and generators.
- – Rotational kinematics and dynamics.
- – Torque and rotational energy.
- These concepts are related to each other and are used to describe the rotation of objects. Here’s a brief overview of each:
- – Angular displacement is the angle an object rotates through.
- – Angular speed is the rate at which it rotates.
- – Angular velocity is the speed and direction of rotation.
- – Angular acceleration is the rate at which the angular velocity changes.
2. Representation by graphical methods of uniform and non-uniform angular acceleration.
- Rotational equations are set up in the same way as linear equation and many linear variables have a rotational counterpart so they are found in a similar fashion to linear equations as shown below
- [math] \begin{gather} \text{Linear velocity} = \frac{\text{displacement}}{\text{time}} \qquad
\text{Angular velocity} (\omega) = \frac{\text{angular displacement}}{\text{time}} \\
\omega = \frac{\Delta \theta}{\Delta t} \\[10pt]
\text{Linear acceleration} = \frac{\text{velocity}}{\text{time}} \qquad
\text{Angular acceleration} (\alpha) = \frac{\text{angular velocity}}{\text{time}} \\
\alpha = \frac{\Delta \omega}{\Delta t} \end{gather}[/math] 
Figure 1 Graphically representation about linear velocity, acceleration and displacement- An angular velocity versus time graph will be a straight line when the angular acceleration is homogeneous. If the acceleration is uniform, determining the gradient may be used to determine the angular acceleration from this kind of graph; if not, determining the gradient of a tangent to the graph at a specific position will allow us to determine the acceleration at that location.
Figure 2 Graphically representation about uniform and non-uniform acceleration- When angular acceleration is uniform, a graph of angular displacement against time will show that displacement is proportional to [math] t^2 [/math].
Figure 3 Graphically representation about uniform and non-uniform acceleration- Similarly, to linear motion, rotational motion has several equations for uniform angular acceleration, which are found below table
-
Linear equation Rotational equation [math] v = u + at [/math] [math]ω_2 = ω_1 + at [/math] [math]s = \frac{1}{2} (u+v)t [/math] [math]θ = \frac{1}{2} (ω_1 + ω_2)t [/math] [math]s = ut + \frac{1}{2} at^2 [/math] [math]θ = ω_1 t + \frac{1}{2} at^2 [/math] [math]v^2 = u^2 + 2as [/math] [math]\omega_2^2 = \omega_1^2 + 2 \alpha \theta [/math] -
Where [math] ω_1 [/math] is initial angular velocity and [math] ω_2 [/math] is final angular velocity.
3. Torque and angular acceleration:
- ⇒Torque:
- Torque (τ) is a measure of the rotational force that causes an object to rotate or twist. It’s a vector quantity, measured in units of Newton-meters (N·m) or foot-pounds (ft·lb).
- Calculation:
- [math] \tau = r \times F [/math]
- Where:
- – [math] \tau [/math] = torque
- – r = distance from the axis of rotation to the point where the force is applied (radius)
- – F = force
- – [math] \times [/math] = cross product operator (indicating the direction of torque is perpendicular to both r and F)
- Units:
- – Newton-meters (N·m)
- – Foot-pounds (ft·lb)
- – Inch-pounds (in·lb)
- Direction:
- Torque is a vector quantity, with direction and magnitude. The direction of torque is determined by the right-hand rule:
- – Point your thumb in the direction of the radius (r)
- – Point your fingers in the direction of the force (F)
- – Your palm will face the direction of the torque (τ)
Figure 4 Appling torque- Types of Torque:
- – Static Torque: Torque that causes an object to rotate at a constant speed.
- – Dynamic Torque: Torque that causes an object to change its rotational speed.
- – Twisting Torque: Torque that causes an object to twist or rotate around a fixed axis.
- – Rotational Torque: Torque that causes an object to rotate around a fixed axis.
- Applications:
- – Mechanical Advantage: Torque is used to gain mechanical advantage in machines and mechanisms.
- – Rotational Kinematics: Torque is used to describe the rotation of objects.
- – Energy Transfer: Torque is used to transfer energy between systems.
- – Rotational Dynamics: Torque is used to describe the rotation of objects under the influence of forces.
- ⇒Torque and angular acceleration:
- Torque (τ) and Angular Acceleration (α) are directly proportional.
- [math] \tau = lα [/math]
- Where:
- – [math] \tau [/math] = torque
- – I = moment of inertia (a measure of an object’s resistance to changes in its rotation)
- – α = angular acceleration
- This equation shows that:
- – When torque is applied to an object, it causes angular acceleration.
- – The more torque applied, the greater the angular acceleration.
- – The moment of inertia (I) affects the relationship between torque and angular acceleration.
- Torque causes angular acceleration.
- Angular acceleration is directly proportional to torque.
- Moment of inertia affects the relationship between torque and angular acceleration.
- Some examples to illustrate this relationship:
- – When you apply torque to a door handle, it causes the door to rotate (angular acceleration).
Figure 5 Appling torque on closing door- – In a car engine, torque is converted into angular acceleration, making the wheels rotate.
Figure 6 Torque on wheel- – In a gyroscope, torque is used to maintain its angular velocity (rotation).
Figure 7 A gyroscope, torque is used to maintain its angular velocity