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PHYS1001: Formula Sheet: Vectors: (3-D)

1-D Motion: Average Velocity and Acceleration:

Instantaneous Velocity and Acceleration:

Scalar (Dot) Product:

Vector (Cross) Product: Constant Acceleration:

Vectors: (2-D)

2-D and 3-D Motion:

Projectile Motion: From (0,0) with

Trajectory:

Range:

Flight Time:

Maximum Height:

Constant Acceleration: (x-y plane)

Uniform Circular Motion:

Angular Speed:

A force is conservative if work done is independent of path.

Period:

Potential Energy:

Frequency:

Relative Motion: Primed frame with displacement unprimed frame:

relative to

In 1-D:

Hooke’s Law for an ideal spring:

Work: For SHM, let

, then

Power: Then, taking

Kinetic Energy: Thus the total energy is

Work – Energy Theorem:

, the kinetic energy is

Simple Harmonic Motion:

Resonance: Peak Amplitude:

has solution: with

.

Full-width at Half-max:

Generic Oscillations:

Gravity: where x0 is stable minimum. Damped Harmonic Motion:

has solution (for

) Impulse of Force:

with Linear Momentum: and

.

Driven Harmonic Motion:

has steady-state solution:

Impulse-Momentum Theorem:

where

1-D Elastic 2 Body Collisions:

and

Centre of Mass:

System of Particles:

where is the angular momentum in the centre of mass frame. Rigid Body Rotations (Fixed Axis):

Momentum:

N2:

Kinetic Energy:

Kinetic Energy of Rotation: Here,

, is the K.E. in centre of mass

frame:

.

Torque:

with

.

N2 for Rotation: Angular Momentum: about axis of rotation. N2 for Rotational Motion:

Angular Momentum:

about axis of rotation.

KE in terms of Angular Momentum:

Analogies Between Linear and rotational Motion Linear Motion Rotational Motion Differential: Differential: Velocity:

Angular Velocity:

Acceleration:

Angular Acceleration:

Momentum:

Angular Momentum:

Force:

Torque:

Precession of a Spinning Top:

Impulse:

Angular Impulse:

The torque due to gravity is

Kinetic Energy:

Kinetic Energy:

Work:

Work:

Power:

Power:

Total KE for Rigid Body:

Work-Energy Theorem for Rigid Rotations:

Since

then after a short time

The new angular momentum makes an angle

with the original direction. When the central axis of the spinning top rotates with the pivot point fixed, we say that the axis precesses. The angular velocity of the precession of the angular velocity vector is then