Units, Dimensions & Kinematics

What does dimensionally homogeneous mean?

Dimensions of LHS = dimensions of RHS

Dimensions of each additive term must be the same

What must always be non-dimensional?

Arguments of trig, functions, exponentials and logs. So sin(wt) must have that wt is non-dimensional hence [wt] = [1] so [w] = [T^-1]

How to find dimensions for derivaites and integrals?

If quantity = Q:

• dQ/dt → divide by [T]

• d²Q/dt² → divide by [T²]

• ∫Q dt → multiply by [T]

What is the position vector in cartesian?

r = x(t)i + y(t)j + z(t)k

What is the velocity in cartesian?

v = dr/dt = dx(t)/dt i + dy(t)/dt j + dz(t)/dt k

What is the acceleration in cartesian?

a = dv/dt = d²r/dt² = d²x/dt² i + d²y/dt² j + d²z/dt² k

What is the magnitude of position?

r = |r| = √(x² + y² + z²)

What is the magnitude of velocity?

v = |v| = √((dx/dt)² + (dy/dt)² + (dz/dt)²)

What is the magnitude of acceleration?

a = |a| = √((d²x/dt²)² + (d²y/dt²)² + (d²z/dt²)²)

If r = cos(wt)i + sin(wt)j (circular motion), what is the velocity and acceleration?

• v = -wsin(wt)i + wcos(wt)j

• a = -w²r (points towards centre - centripetal acceleration)

How to recognise circular motion in exams?

Look for:

• sin/cos in position

• constant radius (x² + y² = constant)

• acceleration proportional to -r

What are the suvat formulas for velocity and position (constant acceleration)?

• v = u + at

• x = ut + ½ at²

How to spot constant acceleration in an exam?

• constant force (e.g. gravity only)

• no velocity-dependent force