Kinematics

Rotational Kinematics

Objective

• Describe rotational quantities using vectors.

• Define and discuss angular displacement, angular velocity and angular acceleration.

Kinematics of Rotational Motion

Describes relationship among:

*angular displacement

*angular velocity

*angular acceleration

*time

Axis of Rotation

the straight line through all fixed points of a rotating rigid body around which all other points of the body move in circles

Angular displacement

Defined as “the angle in radians (degrees, revolutions) through which a point or line has been rotated in a specified sense about a specified axis”. It is the angle of the movement of a body in a circular path.

An object travels around a circle10.0 full turns in 2.5 seconds. Calculate the angular displacement, θ in radians.

Note that 1 complete rotation = 360⁰ = 2π radians = 6.28 rd

• Find: Angular displacement (θ) in radians

• Solution: θ = 10.0 turns ( 6.28 rd / turn ) = 62.8 radians.

  
  

Angular velocity

Also known as angular frequency vector, is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.

If an object travels around a circle with an angular displacement of 70.8 radians in 3.0 seconds, what is its average angular velocity ω in (rad/s)?

• ω =  Δθ / Δt  

• Solution: ω =  70.8 rad / 3.0 s = 23.6 rad/s.

  
  

Angular acceleration

the rate of change of angular velocity with time change in angular velocity: the difference between final and initial values of angular velocity.

translational kinematics vs rotational kinematics

Multiple Choice

The angle through which a point, line, or body is rotated in a specific direction and around a specific axis •

• angular velocity

• angular displacement

• radian

• degree

Multiple Choice

A disk rotates with an angular speed $$\omega$$  . What is the linear speed v of a point on the disk located a distance r away from the axis of rotation?

• $$\omega r$$  

• $$\frac{\omega}{r}$$  

• $$\frac{r}{\omega}$$  

• $$\omega^2r$$  

Multiple Choice

Suppose a bicycle wheel of radius R rolls without slipping for a linear distance x. What is the angular displacement the bicycle wheel during this movement?

• $$\theta=\frac{x}{r}$$

• $$\theta=\frac{r}{x}$$

• $$\theta=xr$$

• $$\theta=xr^2$$

Multiple Choice

What does the slope of a $$\theta$$  vs t graph represent?

• The angular acceleration ( $$\alpha$$  )

• The angular velocity ( $$\omega$$  )

• The angular displacement ( $$\Delta\theta$$  )

• The torque ( $$\tau$$  )

Multiple Choice

What does the slope of a $$\omega$$  vs t graph represent?

• The angular acceleration ( $$\alpha$$  )

• The angular velocity ( $$\omega$$  )

• The angular displacement ( $$\Delta\theta$$  )

• The torque ( $$\tau$$  )

Multiple Choice

What does the area of a $$\omega$$  vs t graph represent?

• The angular acceleration ( $$\alpha$$  )

• The angular velocity ( $$\omega$$  )

• The angular displacement ( $$\Delta\theta$$  )

• The torque ( $$\tau$$  )

Multiple Choice

If the translational velocity of an object is 3 m/s, and is located a distance of 2 m from the axis of rotation, what is the object's angular velocity?

• 0.67 rad/s

• 1.5 rad/s

• 5 rad/s

• 6 rad/s