AP Rotational Kinematics Review

AP review: Rotational Motion and Torque

Mrs Wendt
AP Physics

Rigid Body

•A rigid body is a model of a real extended object. – an object or system of particles in which the distances between particles are fixed and remain constant.

•Solids can be considered to be rigid bodies for the purpose of analyzing rotational motion.

•Fluids are not considered to be rigid bodies.

•When we model an extended object as a rigid body, we assume that the object has a nonzero size but the distances between all parts of the object remain the same (the size and shape of the object do not change).

Center Of Mass

•Center of Mass is the average position of all the parts of the system, weighted according to their masses.

•For simple rigid objects with uniform density, the center of mass is located at the centroid.

•For more complicated shapes, we need a more general mathematical definition of the center of mass : it is the unique position at which the weighted position vectors of all the parts of a system sum up to zero.

Rotational Motion and Equilibrium

•Rotational Motion is when an object turns around an axis.

•Rotation: spinning motion of an object about an axis within the object.

•Revolution: spinning motion of an object about an axis outside of the object.

•Axis of Rotation: (1) imaginary straight line about which an object rotates; (2) one of the reference lines used to depict a coordinate system.

•Rotational Equilibrium – the sum of torques on a body is zero

•the body is rotationally at rest or rotates with a constant angular velocity

•S τi = τ 1 + τ 2 + τ 3 + …. = 0

Static Equilibrium

•Static equilibrium refers to any system where the sum of the forces, and torque, on every particle of the system happens to be zero.

•Simply, it is the equilibrium of a system whose parts are at rest. 

•The fundamental and basic condition for static equilibrium is that an object must not be experiencing any type of motion, irrespective of translational or rotational.

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.

Multiple Choice

A disc rotating at 300 rev/min is stopped after 2 minutes. Calculate

a) the angular acceleration

b) angular velocity after 30 s

c) the number of revolutions

• a) -0.26 rad s^-2

  b) 23.62 rad s^-1

  c) 302 revolutions

• a) -0.45 rad s^-2

  b) 20.35 rad s^-1

  c) 202 revolutions

• a) -0.62 rad s^-2

  b) 36.22 rad s^-1

  c) 112 revolutions

• a) -0.34 rad s^-2

  b) 13.62 rad s^-1

  c) 422 revolutions

​Torque

​Lever arm

Multiple Choice

A pirate captain takes the helm and turns the wheel of his ship by applying a force of 20N to a wheel spoke. If he applies the force at a radius of 0.2 m from the axis to the line of axis of rotation, at an angle of 80º to the line of action, what torque does he apply to the wheel?

• 3.94 Nm

• 4.25 Nm

• 3.56 Nm

• 2.25 Nm

Multiple Choice

A mechanic tightens the lugs on a tire by applying a torque of 110 Nm at an angle of 90º to the line of action. What force is applied if the wrench is 0.4 m long?

• 300 N

• 325 N

• 275 N

• 250 N

Moment of Inertia

Open Ended

Calculate the moment of inertia for a hollow sphere with a mass of 10 kg and a radius of 0.2 m.

​Newton's 2nd Law of Rotation

Rotational Kinetic Energy

​Angular Momentum

​Conservation of angular momentum

The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:

Multiple Choice

What does Rotational Inertia depend on?

• Mass

• Mass Distribution

• Both Mass and its Distribution

• Neither Mass nor its distribution

Multiple Choice

Angular Momentum is the product of...

• rotational inertia and linear velocity.

• linear inertia and angular velocity.

• linear momentum and angular velocity

• rotational inertia and angular velocity.

Multiple Choice

What happens to rotational inertia of an object when mass is moved close to axis of rotation?

• Decreases

• Increases

• Stays the Same

Multiple Choice

A fan spins with a constant angular speed of 500 radians/second. How long will it take to cover an angle of 2π radians (one revolution)?

• 0.013 s

• 0.002 s

• 76.9 s

• 6.28 s

Multiple Choice

You apply force to a wrench to tighten a bolt. The wrench breaks in half. Select the best answer for what happens next. Assume the force applied remains constant.

• The torque is now half of what it was, because torque and distance are inversely proportional
• The torque is now half of what it was, because torque and distance are directly proportional
• The torque is now double of what it was, because torque and distance are inversely proportional
• The torque is now double of what it was, because torque and distance are directly proportional

Multiple Choice

Two bugs ride a turntable which is rotating at a constant rate of 4 rad/s. Bug A is at a radius of 4 cm, while bug B is at a radius of 8 cm. What is the velocity of bug B?  

• 4 cm/s
• 32 cm/s
• 2 cm/s
• 1 cm/s

Multiple Choice

A flywheel starts from rest, and has an angular acceleration of 4 rad/s². What will be its angular speed when it has made one revolution? 

• 5.01 rad/s
• 50.27 rad/s
• 2.83 rad/s
• 7.09 rad/s

Multiple Choice

A top with moment of inertia 0.001 kg·m² is spun on a table by applying a torque of 0.01 Nm for two seconds. If the top starts from rest, find the angular velocity of the top.

• 25 rad/s

• 20 rad/s

• 15 rad/s

• 30 rad/s

Multiple Choice

Gina rolls a bowling ball of mass 7 kg and radius 10.9 cm down a lane with a velocity of 6 m/s. Find the rotational kinetic energy of the bowling ball, assuming it does not slip.

• 60.2 J

• 40.5 J

• 59.3 J

• 50.4 J