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Rigid Body Mechanics (DP IB Physics)

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    Define a moment.

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  • Define a moment.

    A moment is the product of a force and the perpendicular distance from a pivot.

    It describes the turning effect of a force around a particular point, usually when the rotation is incomplete.

  • Define a couple.

    A couple is a pair of forces which are equal in size and act parallel to each other but in opposite directions and along different lines of action on either side of the axis of rotation.

  • True or False?

    A couple produces a net linear force.

    False.

    A couple does not produce a net linear force; it produces a turning effect called torque.

  • State the equation for the moment of a force.

    The equation for the moment of a force is: M space equals space F space cross times space d

    Where:

    • M = moment, measured in newton metres (N m)

    • F = force applied, measured in newtons (N)

    • d = perpendicular distance to pivot, measured in metres (m)

  • What is torque?

    Torque is the product of a force and the perpendicular distance from the axis of rotation to its line of action.

    It describes the turning effect of a force around an axis of rotation.

  • State the equation for the torque of a force about an axis.

    The torque of a force about an axis is: tau space equals space F r space sin space theta

    Where:

    • F = force, measured in newtons (N)

    • r = perpendicular distance between the axis of rotation and the line of action of the force, measured in metres (m)

    • theta = angle between the force and axis of rotation, measured in degrees (degree)

  • How is the torque of a couple calculated?

    The torque of a couple is: tau space equals space 2 F r space sin space theta

    Where:

    • F = force applied, measured in newtons (N)

    • r = perpendicular distance between the axis of rotation and the line of action of the force, measured in metres (m)

    • theta = angle between the force and axis of rotation, measured in degrees (degree)

  • What are the properties of the forces that act in a couple?

    Forces that act in a couple are:

    • equal in size

    • opposite in direction

    • perpendicular to the distance between them

  • What is the SI base unit for torque?

    The SI base unit for torque is newton metres (N m).

  • What effect does the angle, theta, between the force and the axis of rotation have on the resultant torque?

    The effect of angle on the resultant torque is:

    • when the angle is 90° (perpendicular to the axis of rotation), the resultant torque is at a maximum

    • as the angle decreases, torque decreases

    • when the angle is 0° (parallel to the axis of rotation), the resultant torque is zero

  • What is rotational equilibrium?

    Rotational equilibrium is when there is no resultant torque acting on a system.

  • Define the principle of moments.

    The principle of moments states that a body is in rotational equilibrium if the sum of the clockwise moments (torques) is equal to the sum of the anticlockwise moments (torques).

  • True or False?

    In rotational equilibrium, the sum of the torques must be zero.

    True.

    In rotational equilibrium, the sum of the torques must be zero.

  • What is the condition for a beam to be in rotational equilibrium?

    If a beam is in rotational equilibrium: total clockwise torque = total anticlockwise torque.

  • Define unbalanced torque.

    Unbalanced torque is when the net resultant torque is not zero, causing angular acceleration.

  • State whether a plank supported at two points is in rotational equilibrium at the point just before it begins to tip.

    A plank supported at two points is in rotational equilibrium at the point just before it begins to tip is in rotational equilibrium.

  • How can you make calculations of resultant torque as easy as possible?

    To make calculations of resultant torque as easy as possible you should take torque above the point where most of the forces are unknown or zero.

  • Define angular displacement.

    Angular displacement is the angle through which a rigid body has rotated from a fixed reference position.

  • What is the unit of angular displacement?

    The unit of angular displacement is radians (or degrees).

  • State the equation for linear displacement in terms of angular displacement.

    The equation for linear displacement in terms of angular displacement is: s space equals space r theta

    Where:

    • s = linear displacement, or arc length, measured in metres (m)

    • r = distance from the axis of rotation, measured in metres (m)

    • theta = angular displacement, measured in radians (degree)

  • Define angular velocity.

    Angular velocity is the rate of change in angular displacement with respect to time.

  • What is the equation for angular velocity?

    The equation for angular velocity is: omega space equals space fraction numerator increment theta over denominator increment t end fraction​.

    Where:

    • omega = angular velocity, measured in radians per second (rad s-1)

    • increment theta = change in angle, measured in degrees (degree)

    • increment t = change in time, measured in seconds (s)

  • True or False?

    Angular velocity is measured in rad s–1.

    True.

    Angular velocity is measured in rad s–1.

  • State the relationship between linear speed and angular speed.

    The relationship between linear speed and angular speed is v space equals space r omega

    Where:

    • v = linear velocity, measured in metres per second (m s-1)

    • r = distance from the axis of rotation, measured in metres (m)

    • omega = angular velocity, measured in radians per second (rad s-1)

  • Define angular acceleration.

    Angular acceleration is the rate of change of angular velocity with time.

  • What is the equation for angular acceleration?

    The equation for angular acceleration is alpha space equals space fraction numerator increment omega over denominator increment t end fraction​

    Where:

    • alpha = angular acceleration, measured in radians per second squared (rad s-2)

    • increment omega = change in angular velocity, measured in radians per second (rad s-1)

    • increment t = change in time, measured in seconds (s)

  • What is the relationship between linear acceleration and angular acceleration?

    The relationship between linear acceleration and angular acceleration is a space equals space r alpha

    Where:

    • a = linear acceleration, measured in metres per second squared (m s-2)

    • r = distance from the axis of rotation, measured in metres (m)

    • alpha = angular acceleration, measured in radians per second squared (rad s-2)

  • How many radians are there in one full circular rotation?

    In one full circular rotation, there are 2 straight pi radians.

  • How many complete rotations does a wheel make if its angular displacement is 26 rad?

    The wheel makes 4 complete rotations.

    • there are 2π radians in 1 rotation

    • number of rotations = fraction numerator increment theta ​ over denominator 2 straight pi end fraction

    • number of rotations = fraction numerator 26 over denominator 2 straight pi end fraction space equals space 4.138 space almost equal to space 4

  • Convert 15 RPM into rad s−1.

    An angular velocity of 15 RPM is equivalent to 1.6 rad s-1.

    • there are 2π radians in 1 rotation

    • there are 60 seconds in one minute

    • angular velocity: omega space equals space 2 straight pi f space equals space fraction numerator 2 straight pi cross times RPM over denominator 60 end fraction

    • angular velocity: omega space equals space fraction numerator 2 straight pi cross times 15 over denominator 60 end fraction space equals space straight pi over 2 space equals 1.6 space rad space straight s to the power of negative 1 end exponent

  • True or False?

    The initial angular velocity in rotational motion is represented by omega subscript i.

    True.

    The initial angular velocity in rotational motion is represented by omega subscript i.

  • What is the rotational equivalent of the equation v space equals space u space plus space a t?

    The rotational equivalent of open parentheses v space equals space u space plus space a t close parentheses is: omega subscript f ​ equals space omega subscript i ​ plus space alpha t

    Where:

    • omega subscript f = final angular velocity, measured in radians per second (rad s-1)

    • omega subscript i = initial angular velocity, measured in radians per second (rad s-1)

    • alpha = angular acceleration, measured in radians per second squared (rad s-2)

    • t = time, measured in seconds (s)

  • What is the rotational equivalent of the equation s space equals space u t space plus space 1 half a t squared?

    The rotational equivalent of open parentheses s space equals space u t space plus space 1 half a t squared close parentheses is: increment theta space equals space omega subscript i t space plus space 1 half alpha t squared

    Where:

    • increment theta = change in angular displacement, measured in radians (rad)

    • omega subscript i = initial angular velocity, measured in radians per second (rad s-1)

    • t = time, measured in seconds (s)

    • alpha = angular acceleration, measured in radians per second squared (rad s-2)

  • What is the rotational equivalent of the equation s space equals space fraction numerator open parentheses u space plus space v close parentheses over denominator 2 end fraction t?

    The rotational equivalent of open parentheses s space equals space fraction numerator open parentheses u space plus space v close parentheses over denominator 2 end fraction t close parentheses is: increment theta space equals space fraction numerator open parentheses omega subscript i space plus space omega subscript f close parentheses over denominator 2 end fraction t

    Where:

    • increment theta = change in angular displacement, measured in radians (rad)

    • omega subscript i = initial angular velocity, measured in radians per second (rad s-1)

    • omega subscript f = final angular velocity, measured in radians per second (rad s-1)

    • t = time, measured in seconds (s)

  • What is the rotational equivalent of the equation v squared space equals space u squared space plus space 2 a s?

    The rotational equivalent of open parentheses v squared space equals space u squared space plus space 2 a s close parentheses is: omega subscript f squared ​ equals space omega subscript i squared ​ plus space 2 alpha increment theta.

    Where:

    • omega subscript f = final angular velocity, measured in radians per second (rad s-1)

    • omega subscript i = initial angular velocity, measured in radians per second (rad s-1)

    • alpha = angular acceleration, measured in radians per second squared (rad s-2)

    • increment theta = change in angular displacement, measured in radians (rad)

  • Define angular acceleration.

    Angular acceleration is the rate of change of angular velocity with time.

  • What is the unit of angular acceleration?

    The unit of angular acceleration is rad s−2.

  • Define inertia.

    Inertia is the resistance to a change of motion or linear acceleration.

  • True or False?

    A larger mass results in greater inertia.

    True.

    The larger the mass an object has, the greater its inertia.

  • Define the term moment of inertia.

    The moment of inertia is the resistance to a change of rotational motion. It depends on the distribution of mass around a chosen axis of rotation.

  • What is the unit of measurement for the moment of inertia?

    The unit of measurement for the moment of inertia is kg m2.

  • True or False?

    The moment of inertia is the same for all orientations of an object relative to the axis of rotation.

    False.

    The moment of inertia of a body can change depending on its orientation relative to the axis of rotation.

  • State the equation for the moment of inertia of a point mass.

    The equation for the moment of inertia of a point mass is space I space equals space m r ².

    Where:

    • m = mass of object, measured in kilograms (kg)

    • r = distance from axis of rotation, measured in metres (m)

  • What factors affect the moment of inertia of an object?

    The moment of inertia is affected by the object's:

    • shape

    • density

    • orientation relative to the axis of rotation

  • How is the total moment of inertia of a system calculated?

    The total moment of inertia of a system is the sum of the moments of inertia of all the point masses in the system.

  • Define Newton's second law for rotation.

    Newton's second law for rotation states that the torque required to give a rotating object a certain angular acceleration depends on its moment of inertia.

  • State the equation for Newton's second law for rotation.

    The equation for Newton's second law for rotation is: tau space equals space I alpha

    Where:

    • tau = torque, measured in Newton metres (N m)

    • I = moment of inertia, measured in kilograms metres squared (kg m2)

    • alpha = angular acceleration, measured in radians per second squared (rad s-2)

  • What is the equivalent rotational variable for force, F?

    The equivalent rotational variable for force, F is torque, tau.

  • What is the equivalent rotational variable for mass, m?

    The equivalent rotational variable for mass, m is the moment of inertia, I.

  • What is the equivalent rotational variable for acceleration, a?

    The equivalent rotational variable for acceleration, a is the moment of inertia, alpha.

  • What two variables are proportional to each other in Newton's second law for rotational motion?

    In Newton's second law for rotational motion, the two variables that are proportional to each other are:

    • tau

    • alpha

  • What is angular momentum?

    Angular momentum is the product of the moment of inertia and angular velocity of a body.

    It is the rotational equivalent of linear momentum,

  • State the equation for angular momentum.

    The equation for angular momentum isspace L space equals space I omega

    Where:

    • I = moment of inertia, measured in kilograms metres squared (kg m2)

    • omega = angular velocity, measured in radians per second (rad s-1)

  • True or False?

    Angular momentum is conserved even if no net torque acts on a system.

    True.

    The angular momentum of a system remains constant even if no net torque acts on the system.

  • What does conservation of angular momentum mean?

    The conservation of angular momentum means that the total angular momentum of a system remains constant unless acted on by a net external torque.

  • How is the angular velocity, omega, related to linear velocity, v, for a rotating object?

    Angular velocity, omega, is related to linear velocity, v, by the equation omega space equals space v over r.

    Where:

    • r = radius, measured in metres (m)

  • State the equation for the angular momentum of a point mass.

    The equation for the angular momentum of a point mass isspace L space equals space m v r

    Where:

    • I = moment of inertia, measured in kilograms metres squared (kg m2)

    • v = velocity, measured in metres per second (m s-1)

    • r = distance from the axis of rotation, measured in metres (m)

  • What is the principle of conservation of angular momentum?

    The principle of conservation of angular momentum states that the angular momentum of a system always remains constant unless a net torque acts on it.

  • Which factors affect the angular momentum of a system?

    The angular momentum of a system depends on the:

    • Moment of inertia, I

    • Angular velocity, omega

  • What is angular impulse?

    Angular impulse is the change in angular momentum produced by a torque acting over a time interval.

  • What is the difference between linear impulse and angular impulse?

    Angular impulse is the rotational equivalent of linear impulse.

    Linear impulse is the change in linear momentum produced by a force acting over a time interval.

    Angular impulse is the change in angular momentum produced by a torque acting over a time interval.

  • True or False?

    The area under a torque-time graph represents angular impulse.

    True.

    The area under a torque-time graph is equal to the angular impulse or the change in angular momentum.

  • State the equation for angular impulse.

    The equation for angular impulse is increment L space equals space tau increment t

    Where:

    • tau = torque, measured in Newton metres (N m)

    • increment t = time interval, measured in seconds (s)

  • True or False?

    A small torque over a long time has the same effect as a large torque over a short time.

    True.

    A small torque acting over a long time can have the same effect as a large torque acting over a short time.

  • What is the unit of angular impulse?

    The unit of angular impulse is kg m2 s-1 or N m s.

  • How is angular impulse related to angular momentum?

    Angular impulse is the change in angular momentum produced by a torque acting over a time interval.

  • State the equation relating torque and the rate of change of angular momentum.

    The equation relating torque and the rate of change of angular momentum is tau space equals space fraction numerator increment L over denominator increment t end fraction

    Where:

    • tau = resultant torque, measured in newton metres (N m)

    • increment L = change in angular momentum, measured in kilogram metres squared per second (kg m2 s-1)

    • increment t = time interval, measured in seconds (s)

  • What is rotational kinetic energy?

    Rotational kinetic energy is the energy a rotating object possesses due to its angular velocity.

  • State the equation for rotational kinetic energy.

    The equation for rotational kinetic energy is: E subscript k ​ equals space 1 half I omega squared,

    Where:

    • I = moment of inertia, measured in kilograms metres squared (kg m2)

    • omega = angular velocity, measured in radians per second (rad s-1)

  • True or False?

    Rotational kinetic energy depends on linear velocity.

    False.

    Rotational kinetic energy depends on angular velocity.

  • What does it mean when an object rolls without slipping?

    Rolling without slipping is when an object rolls such that its point of contact with the surface has zero velocity relative to the surface.

  • True or False?

    The total kinetic energy of a rolling object includes both translational and rotational kinetic energy.

    True.

    The total kinetic energy of a rolling object is the sum of its translational and rotational kinetic energy.

  • What is the equation for the total kinetic energy of a rolling object?

    The equation for the total kinetic energy of a rolling object is E subscript K t o t a l end subscript ​ equals space 1 half m v squared space plus space 1 half I omega squared

    Where:

    • m = mass, measured in kilograms (kg)

    • v = linear velocity, measured in metres per second (m s-1)

    • I = moment of inertia, measured in kilograms metres squared (kg m2)

    • omega = angular velocity, measured in radians per second (rad s-1)

  • What other forms of energy is gravitational potential energy converted into as an object rolls down a slope?

    Gravitational potential energy is converted into translational and rotational kinetic energy as an object rolls down a slope.

  • What is the moment of inertia for a solid sphere?

    The moment of inertia for a solid sphere is I equals 5 over 2 ​ m r squared

    Where:

    • m = mass of object, measured in kilograms (kg)

    • r = radius of the sphere, measured in metres (m)

Previous:Work, Energy & PowerNext:Galilean & Special Relativity

    1. Topic 1

      Kinematics

    2. Topic 2

      Forces & Momentum

    3. Topic 3

      Work, Energy & Power

    4. Topic 4

      Rigid Body Mechanics

    5. Topic 5

      Galilean & Special Relativity

    1. Topic 1

      Thermal Energy Transfers

    2. Topic 2

      Greenhouse Effect

    3. Topic 3

      Gas Laws

    4. Topic 4

      Thermodynamics

    5. Topic 5

      Current & Circuits

    1. Topic 1

      Simple Harmonic Motion

    2. Topic 2

      Wave Model

    3. Topic 3

      Wave Phenomena

    4. Topic 4

      Standing Waves & Resonance

    5. Topic 5

      Doppler Effect

    1. Topic 1

      Gravitational Fields

    2. Topic 2

      Electric & Magnetic Fields

    3. Topic 3

      Motion in Electromagnetic Fields

    4. Topic 4

      Induction

    1. Topic 1

      Structure of the Atom

    2. Topic 2

      Quantum Physics

    3. Topic 3

      Radioactive Decay

    4. Topic 4

      Fission

    5. Topic 5

      Fusion & Stars

    1. Topic 1

      Measurements in Physics

    2. Topic 2

      Processing Uncertainties

    3. Topic 3

      Scalars & Vectors