PHYSICS 1D03 MIDTERM 2 NOTES
AND SUMMARY
Nicely Written & Complete Study Guide
KEY TOPICS COVERED
• Rotational Kinematics: Angular Position, Angular Velocity, Angular Acceleration, Constant
Angular Acceleration Equations, Linear-Rotational Links
• Torque Dynamics: Unbalanced Torque, Lever Arms, Vector Cross Products, Cartesian
Cross Products, Right-Hand Rule Guidelines
• Rotational Inertia & Dynamics: Moment of Inertia, Mass Distribution, Parallel Axis
Theorem, Newton's Second Law for Rotation
• Center of Mass & Equilibrium: Coordinate Center Calculations, Symmetry Constraints,
Center of Gravity, Static Equilibrium Conditions, Multi-Pivot System Formulations
• Work, Energy, and Power: Vector Dot Products, Work by Constant and Variable Forces,
Work-Kinetic Energy Theorem, Mechanical Power (Translational & Rotational)
• Potential Energy & Conservation: Conservative vs. Non-Conservative Forces,
Gravitational and Elastic Potential Energies, Mechanical Energy Conservation
1
, Lesson 1: Rotational Kinematics of Rigid Bodies
Angular Motion Parameters
A rigid body is an extended object whose constituent particles maintain completely fixed distances relative
to one another during motion. When a rigid body undergoes pure rotation about a fixed axis, every
individual particle travels in a circular path centered on that axis, completing a full revolution in the exact
same time interval.
• Angular Position θ: Measured in radians (rad), where 2π rad = 360° = 1 revolution. The arc length s
along a circle of radius r is: s = rθ.
• Angular Velocity ω: The time derivative of angular position, measured in radians per second (rad/s): ω
= dθ / dt.
• Angular Acceleration α: The time derivative of angular velocity, measured in radians per second
squared (rad/s2): α = dω / dt = d2θ / dt2.
• Directionality: Defined by the right-hand rule. Curl the fingers of your right hand in the direction of
rotation; your extended thumb points along the axis of rotation to indicate the direction of the angular
velocity vector ω. Counter-clockwise (CCW) rotation is standardly defined as positive (ω > 0).
EXAMPLE: LINEAR TRACKING ON SPINNING MEDIA
A compact disc operates with a uniform linear reading velocity of v = 1.3 m/s. To find the angular velocity ω
at the innermost track where the radius is r = 23 mm = 0.023 m, use the connection formula: v = rω.
Rearranging gives: ω = v / r = 1..023 = 56.5 rad/s.
Formulations for Constant Angular Acceleration
When a rigid body undergoes rotation with a uniform angular acceleration (α = constant), its motion is
governed by equations that parallel the standard formulas for linear kinematics:
ωf = ωi + αt
θf = θi + ωit + ½αt2
ωf2 = ωi2 + 2α(θf − θi)
EXAMPLE: DISK SPIN-DOWN REVOLUTION SCALING
A spinning high-speed disc starts from rest and accelerates uniformly to a final target velocity of 7200 rev/
min over an interval of 18 s. First, convert the angular velocity to standard units: ωf = (7200 × 2π) / 60 =
240π rad/s. The constant acceleration is: α = ωf / t = 240π / 18 = 41.89 rad/s2. The net angular displacement
is: Δθ = ½ωft = ½(240π)(18) = 2160π rad. Dividing by 2π yields exactly 1080 revolutions.
2
AND SUMMARY
Nicely Written & Complete Study Guide
KEY TOPICS COVERED
• Rotational Kinematics: Angular Position, Angular Velocity, Angular Acceleration, Constant
Angular Acceleration Equations, Linear-Rotational Links
• Torque Dynamics: Unbalanced Torque, Lever Arms, Vector Cross Products, Cartesian
Cross Products, Right-Hand Rule Guidelines
• Rotational Inertia & Dynamics: Moment of Inertia, Mass Distribution, Parallel Axis
Theorem, Newton's Second Law for Rotation
• Center of Mass & Equilibrium: Coordinate Center Calculations, Symmetry Constraints,
Center of Gravity, Static Equilibrium Conditions, Multi-Pivot System Formulations
• Work, Energy, and Power: Vector Dot Products, Work by Constant and Variable Forces,
Work-Kinetic Energy Theorem, Mechanical Power (Translational & Rotational)
• Potential Energy & Conservation: Conservative vs. Non-Conservative Forces,
Gravitational and Elastic Potential Energies, Mechanical Energy Conservation
1
, Lesson 1: Rotational Kinematics of Rigid Bodies
Angular Motion Parameters
A rigid body is an extended object whose constituent particles maintain completely fixed distances relative
to one another during motion. When a rigid body undergoes pure rotation about a fixed axis, every
individual particle travels in a circular path centered on that axis, completing a full revolution in the exact
same time interval.
• Angular Position θ: Measured in radians (rad), where 2π rad = 360° = 1 revolution. The arc length s
along a circle of radius r is: s = rθ.
• Angular Velocity ω: The time derivative of angular position, measured in radians per second (rad/s): ω
= dθ / dt.
• Angular Acceleration α: The time derivative of angular velocity, measured in radians per second
squared (rad/s2): α = dω / dt = d2θ / dt2.
• Directionality: Defined by the right-hand rule. Curl the fingers of your right hand in the direction of
rotation; your extended thumb points along the axis of rotation to indicate the direction of the angular
velocity vector ω. Counter-clockwise (CCW) rotation is standardly defined as positive (ω > 0).
EXAMPLE: LINEAR TRACKING ON SPINNING MEDIA
A compact disc operates with a uniform linear reading velocity of v = 1.3 m/s. To find the angular velocity ω
at the innermost track where the radius is r = 23 mm = 0.023 m, use the connection formula: v = rω.
Rearranging gives: ω = v / r = 1..023 = 56.5 rad/s.
Formulations for Constant Angular Acceleration
When a rigid body undergoes rotation with a uniform angular acceleration (α = constant), its motion is
governed by equations that parallel the standard formulas for linear kinematics:
ωf = ωi + αt
θf = θi + ωit + ½αt2
ωf2 = ωi2 + 2α(θf − θi)
EXAMPLE: DISK SPIN-DOWN REVOLUTION SCALING
A spinning high-speed disc starts from rest and accelerates uniformly to a final target velocity of 7200 rev/
min over an interval of 18 s. First, convert the angular velocity to standard units: ωf = (7200 × 2π) / 60 =
240π rad/s. The constant acceleration is: α = ωf / t = 240π / 18 = 41.89 rad/s2. The net angular displacement
is: Δθ = ½ωft = ½(240π)(18) = 2160π rad. Dividing by 2π yields exactly 1080 revolutions.
2
