02 - Mechanics
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2.1 Kinematical concepts

2.1.1 : Displacement is a measure of an objects distance and direction from an origin point (it's a vector). Velocity is a measure of the change in displacement over change in time (so it's also a vector) and acceleration is change in velocity over change in time (again a vector)...

2.1.2 : Objects motions will obviously different from different 'positions' or frames of reference. For example, to someone standing on a train going through a station, the station is moving towards the back of the train at, say 10 m/s while to someone on the platform, the train is moving towards the from of the station at 10 m/s...'relative velocity' is the velocity of an object from a given frame of reference...velocity is usually taken from the surface of the earth.

2.1.3 : Velocity is the derivative with respect to time of displacement, and acceleration is the second derivative. Thus, the area under an acceleration graph is velocity, and under velocity is displacement. The graph of velocity will be the graph of the slope of displacement at each point...basically you need to be able to convert between these different types of graphs, or produce them from a written description of an object's motion.

2.1.4 : Instantaneous velocity is the slope of a displacement graph at a point, while average velocity is the slope between two points...ditto for acceleration with the slopes of the velocity graphs.

2.1.5 : Based on a displacement time graph, the average velocity can be found by taking the start and end points, drawing a line between them, and then finding the slope of the resulting line. Instantaneous velocity is the same except the line is the tangent to the curve at the point in question. The units will always be m/s

2.1.6 : Based on a velocity time graph, the average acceleration can be found by taking the start and end points, drawing a line between them, and then finding the slope of the resulting line. Instantaneous acceleration is the same except the line is the tangent to the curve at the point in question. Finding displacement is done by finding the area under the graph (but if the graph goes below the x-axis, the area is still positive).

2.2 Linear motion with constant acceleration

2.2.1 : Derivations of constant acceleration equations...

v = u + at ... this is the basic one...the slope of a velocity graph...a = ^{(v - u)}/_t, which rearranges to the given equation.

The area under a velocity graph is equal to displacement...assuming constant acceleration (otherwise we'd need to use calculus) the area will be a triangle over a rectangle...thus we have the rectangle (base x height) + the area of the triangle (¹/₂ base x height)...the rectangle it ut and the triangle is 1/2 x (v - u) x t but as above, (v - u) = at...so we get the equation s = ut + 1/2at². (This makes more sense if you just integrate wrt t assuming a to be a constant)

Since distance traveled = time x average velocity ... s = ¹/₂(v + u)t

The third and the first equation can then be combined to make v² = u² + 2as....

These are all relatively simple to apply...just plug in the appropriate numbers.

2.2.2 : Calculate acceleration and velocity from strobe photos, light gates etc ... the easiest way is to draw a rough graph, but it can be done by using the above equations...finding the time between each point, and the distance, then finding the velocity...for each point, then the change between the velocities.

2.2.3 : Ignoring air resistance, all objects fall under earth's gravity at the same rate...9.81 m/s². This will not effect horizontal velocity, so this will usually remain constant.

2.3 Concepts of force and mass

2.3.1 : Force is a 'thing' which causes objects to either change their velocity, or change their shape (though physics isn't generally concerned with the latter). Force is a vector, and thus can be resolved into vertical and horizontal...or whatever...force components.

2.3.2 : A free body diagram illustrates all the forces acting on an object...these are usually gravity, friction (though this is ignored in frictionless systems)...objects on something have a force opposing gravity from whatever they're on...and then any forces introduced by the question. The direction of the forces should be shown, but it's usually not necessary to scale the lines for magnitude.

2.3.3 : Mass is defined as a unit of 'the amount of matter'. The SI unit of mass is the kilogram. Weight is a measure of force, specifically the force mass causes in the gravity of the earth. The SI unit is the newton. Newton's second law explains the relationship ... weight = mass x gravitational acceleration ... where gravitational acceleration = 9.81 m/s² on the earth's surface, but will be different on other planets...in lifts accelerating up/down etc...

2.4 Newton's first law of motion

2.4.1 : Newton's first law is basically that objects continue in constant motion (including zero motion) if no force acts on them. Thus, an object traveling at constant speed has no net force acting on it.

2.4.2 : When the net force on an object is zero, it will continue in it's current state of motion...ie translational equilibrium.

2.5 Newton's second law of motion

2.5.1 : The resultant force on a free bond diagram can be found by vector addition. Usually most of the vectors will cancel out, making things simpler.

2.5.2 : mass = ^force/_acceleration. This is Newton's second law, though it is more commonly stated F = ma.

2.5.3 : Whenever an object is accelerating, there must be an unbalanced force acting on it This equation can be used to calculate forces or masses etc based on other information...Examples include elevators (remember they have to be accelerating...otherwise the force of gravity is just being balanced) and various objects being pushed or pulled around...all fairly basic.

2.6 Newton's third law of motion

2.6.1 : Newton's third law...when a particle A exerts a force of B, B exerts an equal magnitude force on A in the opposite sense.

2.6.2 : Applications of Newton's third law to life -- examples ... If you throw a ball at the earth, the ball exerts a force on the ground (earth).Therefore, the earth exerts an opposite force on the ball, making it bounce up...and the earth move down, though only very slightly...cause it has such a big mass. The same thing goes for the gravity exerted on the ball by the earth...the ball also pulls the earth towards it, but again only a little due to the big mass difference.

2.7 : Projectile, simple harmonic and uniform circular motion

2.7.1 : If you throw a ball horizontally off a cliff, it accelerates down, but travels horizontally at constant speed...the two components are independent.

2.7.2 : Problems like the one above must be solved...basically, ignore the horizontal motion...this allows the time to fall to ground to be calculated, and the vertical velocity at the end to be found. Then calculate the distance traveled horizontally in that time, and recompose the vector components to find the final velocity. These calculations can also be done with energy calculations, though it's much more difficult to get directions (necessary for vectors such as velocity).

2.7.3 : An object traveling at constant horizontal speed and accelerating vertically under gravity follows a parabolic path...remember it :)

2.7.4 : Simple harmonic motion ... The classic example of SHM is a pendulum, though it applies to anything where the force bringing the system back to equilibrium is proportional to its extension from it (ie F = -kx ). In simple harmonic motion, the period of the motion is always constant and does not depend on extension.

2.7.5 : Acceleration -- max at extreme extension, minimum at zero extension. Velocity -- max at zero extension, minimum at extreme extension.

2.7.6 : If an object is moving in a circle, then the direction it is moving is constantly changing (the velocity is always at a tangent to the circle). As a result, this change in direction, and thus a change in velocity (even though speed is constant) means there must be acceleration. This acceleration must be towards the center of the circle ( of you draw two direction vectors from close on the circumfrance of the circle, the line between the two points to the center). Thus, the centripetal force and acceleration both towards the center of the circle.

2.8 Linear momentum

2.8.1 : Momentum = mass x velocity ( or p = mv ) and is in Ns^-1 or Kg m s^-1 also, momentum will be a vector (because it contains velocity) and so will be in the same direction as the velocity. Impulse = force x time (the area under a force vs time graph), and is measured in Ns...Impulse will also be a vector, in the same direction as the force. Impulse is generally used as a measure the damage caused in a collision, as if the impulse is greater, there is more damage.

2.8.2 : Force is the rate of change of linear momentum (alternate phrasing of newton's second law) and with it we can derive the F=ma expression...F = ^delta-P/_delta-T = ^{(Pi -Pf)}/_delta-T = ^{(mv -mu)}/_delta-T = ^(m(v-u))/_delta-T = ^{(m x delta-V)}/_delta-T = ma.

2.8.3 : Momentum is always conserved within closed systems. Kinetic energy is conserved in elastic collisions (E_k = ¹/₂mv² ... but that comes up in the next section)

2.8.4 : Solve one dimensional problems ... Rather simple...calculate the total initial momentum, then equate it with the (expression for) the total final momentum and solve for the unknown. If the collision if elastic, the initial and final kinetic energies must be equated in the same way. Inelastic collisions are those in which kinetic energy is not conserved.

2.9 Work, energy and power

2.9.1 : Work = change in energy ... if there is no change in energy, then no work has been done ... because energy is just an amount (a scalar) work is also a scalar. (Measured in joules)

2.9.2 : Work = force x displacement (this assumes we're not changing height or anything...). There may be cases where the force is acting at an angle to the relevant displacement, and so the parallel component of the force must be found.

2.9.3 : Potential energy (In the sense of lifting something up in a gravitational field) ... E_p = mgh, the ... This is not exactly a property of an object, but rather of a system ... ie a ball in the air and the earth ... ie the system has the potential energy ... but it's usually more useful to think of it as belonging to the object in calculations. (Measured in joules, as are all other forms of energy)

2.9.4 : Change in gravitational potential energy = m x g x delta-H. Elastic potential energy generally relates to compressing or stretching springs (or something similar)...as the spring is compressed or stretched away from it's equilibrium point, the elastic potential increase, and when it is released, this energy is converted into kinetic energy. Elastic energy = ¹/₂ks², ie a constant x the extension (or displacement) squared.

2.9.5 : Kinetic energy = ¹/₂mv² ... obviously this can be calculated by knowing the mass and velocity (the energy is a scalar, not a vector).

2.9.6 : Other forms of energy are ... Chemical, nuclear, electrical and thermal (and some others...ie sound)... energy can be converted between these, and the above forms, and is always conserved, so when energy in one form is lost, it must be gained in another.

2.9.7 : Energy is always conserved ... it can't just disappear, but can change from one form to another. Any calculations will generally be related to the ones above which can actually be calculated, for example in a looping roller coaster E_p + E_k = constant, so as potential energy is lost, kinetic energy must be gained, and vice versa.

2.9.8 : Power is defined as a rate of work ... ie P = ^delta-W/_delta-t, and its units will be joules per second (Js^-1) ... any problems involving this will probably involve plugging numbers into that equation.

Other Notes in this Category

1. 01 - Measurement
2. 02 - Mechanics
3. 03 - Thermal Physics and Properties of Matter
4. 04 - Waves
5. 05 - Electricity and Magnetism
6. 06 - Atomic and Nuclear Physics
7. 08 - Measurement
8. 09 - Mechanics
9. 10 - Thermal Physics and Properties of Matter
10. 11 - Waves
11. 12 - Electricity and Magnetism
12. 13 - Atomic and Nuclear Physics

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