The relationship between displacement, velocity and acceleration is summarised in the following diagram. Given the displacement, to find the velocity, we differentiate. To find the acceleration, we differentiate twice. Given the acceleration, to find the velocity, we integrate. To find the displacement, we integrate twice. Example A particle P moves on the– axis. At time […]

# Archive | M2

## Finding the Angle a Hanging Lamina Makes With The Vertical

The centre of gravity of a hanging body always hangs vertically below the point of suspension. If we know the distance of the centre of gravity from two axes, we can use the rules of trigonometry to find the angle a line drawn in the lamina makes with the vertical. The lamina is hung from […]

## Restitution, Momentum, Collisions

When two particles collide, of course we know that momentum is conserved. On top of this, particles also obey Newton’s Law of Restitution: where is the coefficient of restitution. The coefficient is taken to be a fixed constant for collisions between any two bodies. We can then write down two equations from the law of […]

## Framework Hinged to a Wall

A framework hinged to a wall has no symmetry in general. Each force must be calculated individually. To find the tension or compression is each rod for the framework shown below, start by assuming each rod is in tension, so each rod will have a force directed towards its centre. Resolve vertically at C. Resolve […]

## Centre of Mass of a Framework

A framework is a rigid body consisting of a number of rods or wires joined together. As with a lamina formed of different sections, we can find the centre of mass or the famework by equating the moment of the framework about separate axes to sum sum of the moments of the consistuent parts. Consider […]

## Climbing a Ladder Safely II

Unfortunately, a ladder is most likely to slip when the climber is at the top. This is because the moment of the climber in the direction in which the ladder would slip is at its maximum when the climber is at the top. We can analyse the forces on a ladder as it leans against […]

## Successive Impacts

We can analyse the impact between two particles using the Law of Conservation of Momentum and Newton’s Law of Restitution. One of the particles may then go on to collide we another particle and we can again use the same laws. When th particles all lie in the same staright line, analysis becomes especially easy. […]

## Collisions With Walls

When a ball strikes a wall at right angles with a speedit will rebound with a speedwhereis the coefficient of restitution between the ball and the wall. In general however the ball will not strike the wall at right angles, but at some glancing angle, as shown below. In this case the speed of the […]

## Plank Resting on Two Supports

A plank resting on two supports, as below, will experience reaction forcesandat each support. Resolving vertically, (upwards forces equal downwards forces), gives(1) Since we have two unknownsandassuming W is known, besides the equation above, we need one other equation to findandWe can obtain this equation by taking moments, either aboutorTaking moments aboutgivesand from this we […]

## Bodies on Slopes – Toppling

A body will topple when the vertical through the centre of gravity lies just outside the base. A particle on a slope is typically more susceptible to toppling because the vertical is closer to the point where it would lie just outside the base. The triangular prism above is on a slope which makes an […]