**Vector Mathematics**

You will come across vectors in physics problem very frequently.So it is must to know to solve the vector mathematics in short time.And Making sure you have done in correctly.You will find vectors in every module be it mechanics ,electrostatics,magentics

**Addition or Substraction**

When you need to add or substract two or more vectors, you can use following procedure

1. Select a coordinate system that is convenient. (Try to reduce the number of

components you need to find by choosing axes that line up with as many

vectors as possible.)

2. Draw a labeled sketch of the vectors described in the problem.

3. Find the x and y components of all vectors

4. Compute the resultants in x and y direction

5. If necessary, use the Pythagorean theorem to find the magnitude of the resultant vector and select a suitable trigonometric function to find the angle

that the resultant vector makes with the x axis.

**Products**

When u need to perform scalar or vector product.It is nice to following procedure

1. Select a coordinate system(x,y,z) that is convenient. (Try to reduce the number of

components you need to find by choosing axes that line up with as many

vectors as possible.).Use i,j,k as the unit vector across x,y,z axis.

2. Draw a labeled sketch of the vectors described in the problem.

3. Express all the vector in form in i,j,k forms

4. Perform the multipication

**Motion in a Two dimensional Plane**

- Select a coordinate system and resolve the initial velocity vector into x and y

components.

-Find out acceleration in each direction and solve in each direction according to one rectilinear motion equation.

-if the acceleration is in vertical direction only.Follow the techniques for solving constant-velocity problems to analyze the horizontal motion. Follow the techniques for solving constant-acceleration problems to analyze the vertical motion. The x and y motions share the same time of flight t.

- There might be question about trajortory in the Problem ,find out the motion in x and y direction with respect to time from previous point.And then find the value of t from one equation and then put that value in another equation to find out the equation of trajactory

**Motion in a Three dimensional Plane**

- Select a coordinate system and resolve the initial velocity vector into x , y and z components

-Find out acceleration in each direction and solve in each direction according to one rectilinear motion equation.

**Uniform Circular Motion**

-Draw a simple, neat diagram of the system.

-Firstly consider the origin of the forces acting on the each object.To do this find out the field forces acting on the each object.Wherever contact in available account the contact force carefully

-Find out the force acting on the body.The resultant force should provide the required centrepatal required for Circular motion

-Centrepatal

**force=mv**will give the velotiy accordingly.

^{2}/R**Newtons Law Problem**

-Draw a neat diagram of the system.

-Firstly consider the origin of the forces acting on the each object.To do this find out the field forces acting on the each object.Wherever contact in available account the contact force carefully

-Isolate the body whose motion is to be analyzed. Draw a free-body diagram for this body. For systems containing more than one objects, draw separate free-body diagrams for each objects This way there will not any confusion about the force acting on each object.Newtons Third law will help in obtaining the action reaction pair. Do not include in the free-body diagram forces exerted by the object on its surroundings. Establish convenient coordinate axes for each objects and find the components of the forces along these axes.

- Use pseudo force if viewing from the non intertial frame of refrence

-Apply Newton's second law,

**F= ma**, in component form.

-Solve the component equations for the unknowns. Remember that you must have as many independent equations as you have unknowns to obtain acomplete solution.

-Make sure your results are consistent with the free-body diagram.

- If the object are attached to each others by the strings,then make use of constraints theory to find out the acceleration equation between the objects

**Work And Energy**

- Choose your frame of refrence.KE will differ in each refrence frame while PE will remains constant

-Define your system, which may include two or more interacting particles, as well as springs or other systems in which elastic potential energy can bestored. Choose the initial and final points.

-Identify zero points for potential energy (both gravitational and spring). If there is more than one conservative force, write an expression for the potential energy associated with each force.

-Determine whether any nonconservative forces are present. Remember that if friction or air resistance is present, mechanical energy is not conserved.

-Determine whether any external forces are present. Remember that external forces are present, mechanical energy is not conserved.

-If mechanical energy is conserved, you can write the total initial energy at some point

**E**Then, write an expression for the total final energy at the final point that is of interest

_{i}=PE_{i}+KE_{i}.**E**Because mechanical

_{f}=PE_{f}+KE_{f}.energy is conserved, you can equate the two total energies and solve for the quantity that is unknown.

-If frictional forces are present (and thus mechanical energy is not conserved), first write expressions for the total initial and total final energies. In this case, the difference between the total final mechanical energy and the total initial mechanical energy equals the change in mechanical energy in the system

due to friction.

-If external forces are present (and thus mechanical energy is not conserved), first write expressions for the total initial and total final energies. In this case, the difference between the total final mechanical energy and the total initial mechanical energy equals the change in mechanical energy in the system

due to external force.

-If both external forces and frictional forces are present (and thus mechanical energy is not conserved), first write expressions for the total initial and total final energies. In this case, the difference between the total final mechanical energy and the total initial mechanical energy equals the change in mechanical energy in the system due to external force and friction

**Momentum and Collisions**

- Choose your frame of refrence

-Set up a coordinate system and define your velocities with respect to that system.It is usually convenient to have the x axis coincide with one of the initial velocities.

-In your sketch of the coordinate system, draw and label all velocity vectors and include all the given information.

-Write expressions for the x and y components of the momentum of each object before and after the collision. Remember to include the appropriate signs for the components of the velocity vectors.

-Write expressions for the total momentum in the x direction before and after

the collision and equate the two. Repeat this procedure for the total momentum

in the y direction. These steps follow from the fact that, because the momentum of the system is conserved in any collision(

**law on conservation of linear momentum**), the total momentum along any direction must also be constant. Remember, it is the momentum of the system that is constant, not the momentan of the individual objects.

-If the collision is inelastic, kinetic energy is not conserved, and additional information is probably required. If the collision is perfectly inelastic, the final velocities of the two objects are equal. Solve the momentum equations for the unknown quantities.

-If the collision is elastic, kinetic energy is conserved, and you can equate the

total kinetic energy before the collision to the total kinetic energy after the

collision to get an additional relationship between the velocities.And you can solve the energy and momentum equation to find out the find velocities.

- Centre of mass can be useful feature in solving the momentum problems

**Body in Static equilibrium**

-Draw a neat diagram of the system.

-Firstly consider the origin of the forces acting on the each object.To do this find out the field forces acting on the each object.Wherever contact in available account the contact force carefully

-Isolate the object being analyzed. Draw a free-body diagram and then show

and label all external forces acting on the object, indicating where those forces are applied.This way there will not any confusion about the force acting on each object.Newtons Third law will help in obtaining the action reaction pair. Do not include forces exerted by the object on its surroundings.(For systems that contain more than one object, draw a separate free-body diagram for each one.) Try to guess the correct direction for each force. If the direction you select leads to a negative force, do not be alarmed; this merely means that the direction of the force is the opposite of what you guessed.

-Establish a convenient coordinate system for the object and find the components of the forces along the two axes. Then apply the first condition for equilibrium. Remember to keep track of the signs of all force components.

-Choose a convenient axis for calculating the net torque on the object. Remember that the choice of origin for the torque equation is arbitrary; therefore, choose an origin that simplifies your calculation as much as possible.

Note that a force that acts along a line passing through the point chosen as

the origin gives zero contribution to the torque and thus can be ignored.