Models, vectors and moments in mechanics

Mathematical models in mechanics

A mathematical model replaces a real situation with a simpler one that can be solved. Mechanics models make assumptions and state them in standard words. Know what each word lets you do.

Modelling comments

Air resistance is ignored. In reality it opposes the motion, so the real acceleration would be smaller than $g$ and the real speed at the bottom would be less than the model predicts.

Vectors in mechanics

Displacement, velocity, acceleration and force are vectors. In a plane, write them using unit vectors $\mathbf{i}$ and $\mathbf{j}$ in two perpendicular directions, or as column vectors.

$\mathbf{v} = 3\mathbf{i} - 4\mathbf{j} \quad \text{or} \quad \mathbf{v} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$

Magnitude and direction

For $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$:

$|\mathbf{v}| = \sqrt{a^2 + b^2}, \qquad \tan\theta = \frac{b}{a}$

where $\theta$ is the angle with the $\mathbf{i}$ direction. Draw a sketch: the signs of $a$ and $b$ tell you the quadrant.

Speed $= \sqrt{3^2 + 4^2} = 5$ m s$^{-1}$. The angle below $\mathbf{i}$ is $\tan^{-1}\dfrac{4}{3} = 53.1°$.

Resultant forces and Newton's second law

The resultant of several forces is their vector sum. For constant mass:

$\mathbf{F} = m\mathbf{a}$

Resultant $= (6\mathbf{i} + 4\mathbf{j})$ N, so $\mathbf{a} = (3\mathbf{i} + 2\mathbf{j})$ m s$^{-2}$.

Constant acceleration in vector form

$\mathbf{v} = \mathbf{u} + \mathbf{a}t, \qquad \mathbf{s} = \mathbf{u}t + \tfrac{1}{2}\mathbf{a}t^2$

$\mathbf{s} = 4(2\mathbf{i} + \mathbf{j}) + \tfrac{1}{2}(16)(\mathbf{i} - 2\mathbf{j}) = (8\mathbf{i} + 4\mathbf{j}) + (8\mathbf{i} - 16\mathbf{j}) = (16\mathbf{i} - 12\mathbf{j}) \text{ m}$

Position vectors

A particle moving with constant velocity $\mathbf{v}$ from position $\mathbf{r}_0$ has position $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t$. Two particles meet when their position vectors are equal at the same time $t$, so equate the $\mathbf{i}$ components and the $\mathbf{j}$ components.

Moments

The moment of a force about a point is the force multiplied by the perpendicular distance from the point to the line of action of the force.

$\text{moment} = F \times d \quad \text{(units N m)}$

A moment turns clockwise or anticlockwise. Choose one as positive and keep to it.

Equilibrium

A body is in equilibrium when the resultant force is zero and the total moment about any point is zero. In a problem about a rigid body at rest, take moments about a point where an unknown force acts, so that force drops out.

Take moments about $A$. The weight acts at the middle, 2 m from $A$: $20g \times 2 = R_C \times 3$, so $R_C = \dfrac{40g}{3} = 131$ N (using $g = 9.8$).

Moments about $A$: $20g \times 2 + 5g \times 4 = R_C \times 3$, so $R_C = \dfrac{60g}{3} = 20g = 196$ N.

A plank about to tilt

If a plank is on the point of tilting about a support, the reaction at the other support is zero. This gives the limiting position or the largest load.

Non-uniform bodies

For a non-uniform rod, the weight acts at the centre of mass, which is not at the middle. Use the given position or the position you are asked to find.

Take moments about the right end $B$: $R_A \times 3 = 6g \times 2$, so $R_A = 4g = 39.2$ N.