Solving Physics Problems

A common approach to learning physics is to memorize every equation and formula you encounter and to do a zillion different problems in hopes that something similar will appear on a test. Unfortunately, it is a very bad approach for a number of reasons. First, physics instructors such as myself have a fearsome habit of putting problems on test that are totally unfamiliar. Second there always appears to be an overwhelming number of equations - dozens per class! Third it requires enormous amounts of time and effort. Fourth, it convinces many otherwise successful students that physics is hard and very, very boring. If this was the only way to learn physics there wouldn't be too many physicists in the world today.

Luckily there is another way. First, one masters about a dozen concepts such as Vectors, Newton's Laws, Conservation of Energy and so on. There aren't many formulas involved and the ones that do appear are rather simple like F = ma and E_final = E_initial. Next one learns how to apply these techniques to solve problems. Applying the technique means using the physics concept to write the specific equations that apply to the problem of interest. Finally one learns the characteristics of the problems that indicate which technique to use. Of course, you do need plenty of practice to get good at this but unlike the first approach, you aren't working blindly and you acquire confidence as you go along.

Below is a set of crib notes for the various topics we will cover. It points out the clues, or identifying characteristics, in the problem that indicates which concept to use. The method of solution is briefly outlined and a few warning reminders of some tricky points.

Vectors

Characteristic	Displacement, velocity, acceleration, forces, momentum, impulse, electric field, magnetic field
Solution	Break each vector into x and y components Add up x components Add up y components Magnitude of sum is given by Angle in right triangle formed by A, Ax, and Ay is given by θ = tan-1(|Ay/Ax|) State angle in conventional form
Warning	Be careful of signs. Be consistent.

Projectiles

Newton's Second Law

Characteristic	Problem asks about forces or accelerations Problem has an object moving in a circle (see UCM)
Solution	Draw one free body diagram (see Forces) for each object in problem Decide on a direction for the acceleration for each object Choose axes such that one axis points in direction of the acceleration Determine the x and y components of each force Get an equation for the x components for each object using Get an equation for the y components for each object using If there is friction in the problem, write an equation for it using and
Warning	The acceleration and the velocity of an object do not have to be in the same direction! Cannot solve problems with springs unless you know the acceleration Be careful of signs. Be consistent.

Uniform Circular Motion

Satellites

Characteristic	Problem has an object (the satellite) orbiting a much more massive object
Solution	Use the satellite equations, and May need to use v = 2πR/T
Warning	Are all the units SI (m, kg, s)? If not change to SI. R refers to the distance between the centres of the objects

Newton's Law of Gravitation

Characteristic	Problem has an multiple objects in outer space (Type A) Problem asks about acceleration due to gravity on another planet (Type B)
Solution	Type A Use which is always attractive and a vector Follow Newton's Second Law Type B Use
Warning	Are all the units SI (m, kg, s)? If not change to SI. R refers to the distance between the centres of the objects

Centre of Mass

Characteristic	Problem has an object with a complex shape in a static equilibrium problem (Type A) Problem has a system which changes shape under only internal forces (Type B)
Solution	Type A For a uniform simple shape, CM is at geometric centre The CM of a complex shape is determined from the CM's of its pieces Treat holes as objects of negative mass For uniform object, the mass of each piece or hole is proportional to area or volume of the whole Use and Type B The centre of mass cannot move Write the location of CM of each piece of system, before and after, in the same coordinate system Equate xcm for both cases Equate ycm for both cases
Warning

Moment of Inertia

Characteristic	Problem has an object with a complex shape in a rotational dynamics problem
Solution	Moment of Inertia is the sum of the moments of each piece about the same axis, Itotal = I1 + I2 + … If the axis of rotation is not the through the centre of mass of the piece, Ipiece = ICM + Mpieced2 ICM is given in table for most shapes Treat holes as objects of negative mass/moment For uniform object, the mass of each piece or hole is proportional to area or volume of the whole Rods are cylinders rotating perpendicularly Moment of inertia for one or more particles is
Warning	Only use ICM for the parallel axis theorem

Torque, Static Equilibrium, Rotational Dynamics

Characteristic	Problem has an object that might, or does, rotate Problem asks about angular acceleration Problem asks about a torque Problem has a rolling object
Solution	Draw one extended free body diagram (see Forces) for each object in problem Follow Newton's Second Law Decide on a direction for the angular acceleration for each object Choose a single point to from which to determine all torques and Moments of Inertia Determine the torques caused by each force Use or Note CCW is +, CW is - Tension in rope is different on either side of a pulley Determine Moment of Inertia (see Moment of Inertia) Complex shapes can be broken into pieces. The mass of each piece is proportional to area or volume for uniform objects
Warning	Be careful of signs. Be consistent.

Work and Energy

Impulse and Collisions

Momentum and Collisions

Angular Momentum

Simple Harmonic Motion

Standing Waves

Characteristic	Problem has a vibrating string or air column Problem asks about a vibrating long thin wire or rod or the like
Solution	For strings may need to use or If the string is fixed at both ends or air column is open at both ends Use Use If the string or air column has one open end and one fixed end Use Use
Warning	Do not confuse the mass of the string, M, with the hanging mass that creates Ftension

Sound Level / Decibels / Sound Intensity

Characteristic	Problem asks about intensity of a noise or sound Problem asks about sound level of noises or sounds Problem mentions that the noises are incoherent
Solution	Incoherent intensities from multiple sources add, Itotal = I1 + I2 + … For a spherical source Sound level is given by Intensity is related to sound level, β, by
Warning	The sounds must be incoherent to use above Solution! Be careful with the log function and powers of 10 Do not confuse an intensity problem with an interference problem

Doppler Shift

Characteristic	Problem asks about the frequency heard by a listener when either source or listener is moving
Solution	Determine whether source is approaching or receding from listener Determine whether listener is approaching or receding from source Use
Warning

Interference

Characteristic	Problem asks where waves from two sources will interfere constructively or destructively
Solution	The waves must be of the same frequency (be coherent) Constructive interference (maximum amplitude) when Destructive interference (zero or minimum amplitude) when
Warning	Δx is the difference in the length of the two paths that the two waves take Do not confuse an intensity problem with an interference problem

Forces

• Near surface of planet, object has weight W = mg
• Weight acts from centre of mass
• Far from planets, use Law of Gravity
  

• There is a normal force for every surface touched
• Normal acts from surface through object perpendicular to surface of contact
• Normal force is the apparent weight (feeling of weight")
• Normal force is zero if and only if object loses contact with surface
  

• There is a frictional force point along the surface
• If the objects slides over surface, there is kinetic or sliding friction
• If the object does not slide relative to the surface, there is static friction
• Kinetic friction is always opposite to the motion of the object
• Static friction may point in either direction. The direction of the static friction is found by asking which way the object would move if a little more or a little less force was applied. Static friction opposes the potential motion
  

• There is a tension force
• Ropes and strings can only pull
• Direction of tension is from object along string

Questions? mike.coombes@kwantlen.ca