Mathematics: Mechanics 3 (Notes 1 of 2)

Newton's \(3^{rd}\) Law states that 'every action has an equal and opposite reaction'.This means that is two ladders meet at a point and A puts a force upwards on B then B will exert an equal force downwards on A.For this reason if a rod is jointed at a point, it can be considered as two separate rods with the horizontal and vertical components of the force at the point being drawn in opposite directions. From Mech 2:The normal reaction is the force which acts at right angles to the surfaces in contact. Friction is always less than or equal to \(\mu\) x R and the relationship \(F_{max}\)=\(\mu R\) should be applied. - there will often be two different coefficients of friction in a problem, for example between a ladder and a wall and between a ladder and the floor.Other methods which can be applied to these problems are resolving in 2 perpendicular directions (normally horizontally and vertically) and taking moments.

The same equations apply to both elastic springs and strings when they have been stretched past their equilibrium position, however once springs reach their natural length they will compress whereas strings will not so the equations would no longer be accurate.Hooke's Law is that the tension, T, in a string or spring = \(\frac{\lambda x}{l}\) where x is the extension and l is the natural length. \(\lambda\) is the modulus of elasticity and is a property of the spring or string. The elastic potential energy can be found by integrating this equation wrt to x (from physics in a T against x graph EPE is the area under the graph). This gives the equation EPE=\(\frac{\lambda x^2}{2l}\).When attempting to find the velocity of a string at a given extension or the extension of a string for a given velocity conservation of mechanical energy can be used:\(PE_{before}\) + \(KE_{before}\) + \(EPE_{before}\) = \(PE_{after}\) + \(KE_{after}\) + \(EPE_{after}\)It is important to take care if using this when considering a string because of the difference in compression properties beyond the natural length.

When dealing with oblique impacts between a sphere and a plane (or two spheres) split the problem into it's horizontal and vertical components to turn it into two '1-dimensional' problems.Momentum = mass x velocity and so, as velocity is a vector quantity, momentum is also a vector quantity. Because Impulse = Change in Momentum then Impulse is also a vector quantity.  Use conservation of linear momentum (parallel to the plane). And e = \(\frac{separation speed}{approach speed}\), impulse = change in momentum (perpendicular to the plane).

Equilibrium of Rigid Bodies in Contact

Elastic Strings and Springs

Impulse and Momentum (2 dimensions)