Particle Dynamics
Particle Dynamics
Note that whereas 'kinematics' attempts to describe motion within explaining it, 'dynamics' attempts to explain motion and is therefore largely the study of FORCES and their effects.
5.1 Inertial mass
- Although mass is usually defined as 'amount of matter' it can also be defined as 'resistance to acceleration'
- Resistance to acceleration, or tendency to maintain a constant velocity (which may or may not be zero) is called 'inertia'
- It is clear from the above two lines that 'inertial mass' is equivalent to 'inertia', thus if inertia were to be given units the units would be kilograms
5.2 Newton’s First Law of Motion – Inertia
Galileo concluded that if it were not for friction, an object in motion would keep moving forever.
Newton's first law of motion (the law of inertia): Every object continues in a state of rest, or in a state of motion in a straight line at constant speed, unless it is compelled to change that state by forces exerted upon it.
- The vector sum of all the forces acting on an object is called the 'net force' or the 'resultant force'.
- When the net force is zero (i.e. all the forces cancel one another) then the forces are said to be balanced.
- Inertia is the resistance an object has to a change in its state of motion i.e. resistance to acceleration. Inertia is not a force.
- The mass of an object is the amount of matter in it. This is equivalent to saying that mass is a measure of inertia. Mass depends only on the number and kind of atoms in the object -
- Mass is not the same as weight and it does not depend on the location of the object.
- The weight of an object is the gravitational force acting on it. Weight depends on the location of the object.
- In the SI system, mass is measured in kilograms and weight is measured in newtons.
- An object at rest, with zero net force acting on it, is said to be in static equilibrium. A moving object, with zero net force acting on it, is said to be in dynamic equilibrium.
- When an object is in static equilibrium, its weight force is balanced by an equal and opposite support force.
- When an object moves with constant velocity while an applied force acts on it, an equal and opposite force, usually friction, must also act to balance the applied force.
5.3 Newton’s Second Law of Motion – Force and Acceleration
- An object accelerates (changes speed and/or direction) when a net force acts on it.
- The acceleration of an object is directly proportional to the net force acting on it.
- The acceleration of an object is inversely proportional to the mass of the object.
- Acceleration equals net force divided by mass. a = F_{net} / m or F_{net} = ma
- The acceleration is in the same direction as the net force.
5.5 The difference between mass and weight
- Weight is the force of gravity acting on an object. This is not the same as mass, since mass is defined as the amount of matter in an object.
- Weight is a force and is measured in newtons (N). Mass is measured in kilograms (kg).
- Weight can be calculated as mass times the acceleration of free fall W = mg
- Thus near the surface of planet earth, where g = 9.8 m/s^{2}, one kilogram has a weight of 9.8N
"Weightlessness" is strictly speaking NOT experienced by orbiting objects such as astronauts since the FORCE OF GRAVITY ACTING ON THEM (their "weight" as defined in science) is far from zero (the weight force is what keeps them moving in a circular orbit instead of moving in a straight line). However they may 'feel' weightless because there is nothing supporting them. They are 'falling' at the same rate as their spacecraft... without getting any closer to the ground. To achieve true weightlessness you would need to be far away from any planet or star, where the gravitational field is zero.
5.6 Newton’s Third Law of Motion – Action and Reaction
- An interaction between two things produces a pair of forces, for each object exerts a force on the other.
- The two interacting forces are called the action force and the reaction force. It makes no difference which force is called the action force.
- Action and reaction forces are equal in strength and opposite in direction.
- The action and reaction forces always act on different objects, unlike the balanced forces of Newton's First Law which always act on the same object.
Newton's Third Law: When object A exerts a force on object B then B exerts an equal and opposite force on A.
Note that the old-fashioned statement of Newton's Third Law "For every action there is an equal and opposite reaction" is less satisfactory since it does not emphasize that the forces act on different objects.
5.7 Normal force (Orthogonal force) and centripetal force
- The normal (orthogonal) force is the perpendicular component of the contact force between two surfaces.
- centripetal force is the net force acting on a object moving in a circle or in an arc - it is a force towards the center of the circle
- F_{cent} = mv² / r
- the outward force known as 'centrifugal force' does not exist! It is an illusion - the impression that there is an outward force is due to the inertia of the moving object.
5.8 Friction
- friction is a force that opposes relative motion - friction is the force that decelerates a car that is slowing down on a flat road, but friction is also the force that accelerates a car that is speeding up on a flat road.
- friction is due to the interlocking of the two surfaces. Even surfaces that may appear smooth to the naked eye would appear rough if viewed through a microscope
- frictional force = f, normal (perpendicular) force = n
- the coefficient of friction, µ, depends only on the roughness of the two surfaces
- µ is the ratio of two forces and therefore has no units
- f_{s} <= µ_{s} n
- f_{s max} = µ_{s} n
- f_{k} <= µ_{k} n
- µ_{s }>_{ }µ_{k}
- the last equation indicates that the maximum value of the static frictional force for two given surfaces is greater than the value of the kinetic frictional force - this has implications for car braking systems and leads to the concept of ABS (antilock braking systems)
- note that for tires rolling along a road, the friction is considered static rather than kinetic since the tires are not sliding
- the above equations are 'empirical' i.e. useful approximations based on experimental results but with no theoretical base
- according to the above equations, the frictional force does NOT depend on the area of contact
- according to the above equations, the force of kinetic friction does NOT depend on the speed that the two surfaces slide past one another
- µ_{s} = tan θ for an object on inclined plane that has been tilted to the maximum possible angle without the object sliding
- µ_{k} = tan θ for an object on inclined plane that has been tilted such that an object that has been set in motion continues to move at a constant velocity
- note that in the previous two lines Newton's first law is applicable
5.9 Solve problems related to Newton's laws including incline planes
- the key to solving inclined plane problems is to resolve the weight force into components parallel and perpendicular to the plane
- component of weight perpendicular to inclined plane = n = mg sin θ
- component of weight parallel to inclined plane = mg cos θ
5.10 Impulse and Momentum
- The momentum (p) of an object is the product of its mass and its velocity p = m v
- In the SI system, the units for momentum are kg m / s
- The change in momentum depends on the force that acts and the length of time it acts.
- Impulse is average force multiplied by the time during which it acts.
- The impulse exerted on something is equal to the change in momentum it produces.
- F t = Δp
The law of conservation of momentum states that if no net external force acts on the system then the total momentum remains the same.
- When objects collide in the absence of external forces, momentum is always conserved no matter whether the collision is elastic ('bouncy') or inelastic ('sticky').
- Kinetic energy is conserved only in perfectly elastic (perfectly 'bouncy') collisions.
Momentum is a vector quantity, so momentum addition is vector addition (diagrams, arrows, components...).