# Archive | M1

## Work Done Against Friction and Gravity

If a particle speeds up then energy must be given to the particle to increase it’s kinetic energy. If it is moving up a slope then some of the energy supplied must go to increase the potential energy of the particle, and if there is any resistive force then some of the supplied energy must […]

## M1 Finding the Minimum Distance Between Two Moving Points

Suppose we have two points. Both points are moving. Up to a certain time they are moving towards each other and afterwards they are moving away from each other. We need to find the minimum distance between the points and the the value of time, t, that gives this minimum distance. r1(t)=3i+4j+t(5i-j) and r2(t)=5i-2j+t(i-3j) In […]

## Two Pulley Systems

If a light string passes over two smooth light pulleys, then the tension in the string will be the same throughout. If one of the pulleys if fixed to the ceiling and the string passes around both pulleys in the arrangement below, we can applyto find the acceleration of each mass and the tension in […]

## Collisions in Mid Air

Particles will collide if they are in the same place at the same time. Particles that collide in mid air, subject only to the force of gravity will be moving as projectiles, and the simplest typ of collision that can take place is between particles moving vertically. Suppose then that one particle is thrown up […]

## Displacement Between Particles Starting to Move at Different Times

If a particle A starting from a pointatmoves with constant velocitythe displacement vector of the particle at timeis given byIf another particle B starts from some other pointa short timeseconds later, we will not be able to use the same equation to find the position vector of particle B. IF THE TWO PARTICLES START AT […]

## Using Vectors to Fond the Centres of Shapes

We can label the sides of polygons with letters that represent vectors. The length of the vector is (typically) the length of the side and the direction of the vector is along the side. We can often use vectors to determine the centre of the shape. Suppose for example that we want to find the […]

## Finding the Weight and Centre of Gravity of Non Uniform Beams

We cannot assume that the centre of gravity of a non uniform beam is in the centre of the beam. If we do not also know the centre of the beam, then we have a problem. Fortunately there is quite a neat solution which enables us to find both the position of the centre of […]

## Acceleration, Velocity and Displacement

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 […]

## Bearings

Imagine you are at a point A and you want to go to a point B. You must travel in the direction that takes you from point A to point B. You can plot the positions of A and B, and find the bearing from A to B by superimposing on A a compass, with […]

## Solving Particle on Slope Problems

Find the acceleration and tension in the string for the system above. We start by drawing all the forces acting on each particle. Resolving Perpendicular to the slope for P: R-3g cos 30=0 rightarrow R=3g cos 30 Applying F=ma for P: T-%mu R -3gsin 30 =3a T-3 %mu gcos 30 -3gsin 30 =3a (1) Applying […]