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A Level Mechanics
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Trigonometry for Mechanics
Find sin θ and cos θ from tan θ
\[ \tan \theta = \frac{3}{4} \rightarrow \sin, \cos \]
Use a right triangle to find sin and cos when tan is given.
Resolve velocity into components
\[ v \rightarrow v_x, v_y \]
Split a velocity into horizontal and vertical parts.
Resolve force into plane components
\[ F \rightarrow F_\parallel, F_\perp \]
Split a force into parts parallel and perpendicular to plane.
SUVAT Equations
v = u + at to find final velocity
\[ v = u + at \]
Calculate final velocity from initial velocity and acceleration.
v = u + at to find time
\[ t = \frac{v – u}{a} \]
Rearrange v = u + at to find time.
s = ut + ½at² to find displacement
\[ s = ut + \frac{1}{2}at^2 \]
Find displacement under constant acceleration.
v² = u² + 2as to find velocity
\[ v^2 = u^2 + 2as \]
Find final velocity when time is not given.
v² = u² + 2as to find displacement
\[ s = \frac{v^2 – u^2}{2a} \]
Find displacement when time is unknown.
s = ½(u + v)t to find displacement
\[ s = \frac{1}{2}(u + v)t \]
Use average velocity formula to find distance.
Speed-Time Graphs
Distance from area under graph
\[ \text{Distance} = \text{Area} \]
Find distance by calculating area under speed-time graph.
Acceleration from gradient
\[ a = \text{gradient} \]
Find acceleration from gradient of speed-time graph.
Deceleration from gradient
\[ \text{dec} = |\text{gradient}| \]
Find deceleration from negative gradient.
Vector Kinematics
Velocity by differentiating position
\[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \]
Differentiate position vector to find velocity.
Acceleration by differentiating velocity
\[ \mathbf{a} = \frac{d\mathbf{v}}{dt} \]
Differentiate velocity vector to find acceleration.
Position by integrating velocity
\[ \mathbf{r} = \int \mathbf{v} \, dt \]
Integrate velocity vector and use initial conditions.
Speed from velocity vector
\[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2} \]
Calculate magnitude of velocity vector.
Direction of motion
\[ \tan \theta = \frac{v_y}{v_x} \]
Find direction from velocity vector.
Time when moving parallel to vector
\[ \mathbf{v} \parallel \mathbf{d} \]
Find when velocity is parallel to given direction.
Constant acceleration vector SUVAT
\[ \mathbf{r} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2 \]
Apply r = ut + ½at² with vectors.
Forces and Newton’s Laws
F = ma to find force
\[ F = ma \]
Calculate force from mass and acceleration.
F = ma to find acceleration
\[ a = \frac{F}{m} \]
Calculate acceleration from force and mass.
Weight and normal reaction
\[ R = W = mg \]
Find normal reaction on horizontal surface.
Resultant of perpendicular forces
\[ R = \sqrt{F_1^2 + F_2^2} \]
Find magnitude of resultant using Pythagoras.
Friction
Maximum friction force
\[ F_{\max} = \mu R \]
Calculate Fmax = μR.
Find coefficient of friction
\[ \mu = \frac{F}{R} \]
Calculate μ from friction and normal reaction.
Acceleration with friction
\[ a = \frac{F – \mu R}{m} \]
Find acceleration when friction opposes motion.
Inclined Planes
Normal reaction on inclined plane
\[ R = mg \cos \alpha \]
Resolve perpendicular to plane to find R.
Component of weight down the plane
\[ W_\parallel = mg \sin \alpha \]
Resolve parallel to plane to find component of weight.
Acceleration down rough inclined plane
\[ a = g(\sin \alpha – \mu \cos \alpha) \]
Apply Newton’s second law on inclined plane with friction.
Projectile Motion
Time to reach maximum height
\[ t = \frac{u}{g} \]
At maximum height, vertical velocity = 0.
Maximum height of projectile
\[ s = \frac{u^2}{2g} \]
Use v² = u² – 2gs with v = 0.
Horizontal range of projectile
\[ R = u_x \times T \]
Calculate horizontal distance travelled.
Velocity at given time
\[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2} \]
Find velocity components at a given time.
Moments
Calculate moment of force
\[ M = F \times d \]
Use moment = force × perpendicular distance.
Find perpendicular distance
\[ d_\perp = L \cos \alpha \]
Find perpendicular distance for moment calculation.
Equilibrium condition for moments
\[ \sum M = 0 \]
Set up and solve equilibrium equations.
Connected Particles
Tension over smooth pulley
\[ T = \frac{2m_1m_2g}{m_1+m_2} \]
Find tension in string over pulley.
Acceleration of connected system
\[ a = \frac{(m_2-m_1)g}{m_1+m_2} \]
Find common acceleration over pulley.
Particle on table connected to hanging mass
\[ a = \frac{m_2g}{m_1+m_2} \]
Set up equations for particle on table with pulley.
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