G484 - The Newtonian World

·         Linear momentum is the product of mass and velocity. Momentum = mass x velocity, p = mv

·         Principle of conservation of momentum: For a closed system, in any specified direction, the total momentum before an interaction (e.g. collision) is equal to the total momentum after the interaction.

·         In all interactions or collisions, momentum and total energy are conserved.

·         Kinetic energy is conserved in a perfectly elastic collision.

·         Kinetic energy is not conserved in an inelastic collision. In such a collision, kinetic energy is transformed into other forms of energy (e.g. heat or sound). Most collisions are inelastic.

Define momentum.

State the principle of conservation of momentum.

What is conserved in all interactions?

What is conserved in a perfectly elastic collision?

Which category do most collisions fall under?

Define a "perfectly elastic" collision.

Define an "inelastic" collision.

Newton’s first law of motion states that “An object will remain at rest or keep travelling at constant velocity unless it is acted on by an external force”. Newton’s second law of motion states that “The net force acting of an object is equal to the rate of change of its momentum. The net force and the change in momentum are in the same direction”. F = m(v-u)/t. Newton’s third law of motion states that “When two bodies interact, the forces they exert on each other are equal and opposite.” The net force acting on a body is equal to the rate of change of its momentum. Net force = rate of change of momentum or F = p/t. The equation F = ma is a special case of Newton’s Second Law of Motion when mass m remains constant.  The impulse of a force is defined as the product of the force F acting on an object and the time t for which it acts. Impulse = force x time or Impulse = F x t. For a varying force, the impulse is equal to the area under the force against time graph. The impulse of a force is equal to the rate of change of momentum of a body. Impulse = change in momentum or Impulse = Δp

State Newton's 1st Law.

State Newton's 2nd Law.

State Newton's 3rd Law.

What is the net force equal to? Equate this.

What makes Newton's 2nd Law a special case?

Define the "impulse" of a force.

What is the impulse equal to for a varying force?

What is the impulse equal to for a constant force?

 Angles can be measured in radians. An angle of 2π radians is equal to 360^o. An object moving at a steady speed along a circular path has uniform circular motion. The angular displacement θ is a measure of the angle through which an object moves in a circle. An object moving in a circle is not in equilibrium, it has net force acting on it. The net force acting on an object moving in a circle is called the centripetal force. This force is directed towards the centre of the circle and is at right angles to the velocity of the object. The magnitude of the centripetal force F acting on an object of mass m moving at a speed v in a circle of radius r is given by F = mv²/r. An object moving in a circle has a centripetal acceleration a given by a = v²/r.

Other equations to use for circular motion are as follows:

ω = 2πf =2π/T, f = 1/T, a = rω^2, s = θ/r,  v = rω, v =2πrf, v = 2πr/T

What units can angles be measured in? State the relationship between the units.

Does an object moving at a steady speed automatically undergo a circular path?

What is the angular displacement?

Are all the forces balanced in a circular object?

Define "centripetal force".

State the equation for centripetal force.

State the equation for centripetal acceleration.

State 5 other equations that can be used for circular motion, and define all terms used.

The force of gravity is an attractive force between any two objects due to their masses. The gravitational field strength g at a point is the gravitational force exerted per unit mass on a small object placed at that point –  that is: g = f/m. The external field of a uniform spherical mass is the same as that of an equal point mass at the centre of the sphere. Newton’s Law of Gravitation states that “Any two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of their separation.”  The equation for Newton’s Law of Gravitation is F = - GMm/r² where G is the gravitational constant given by 6.67 x 10^-11. The gravitational field strength at distance r from a point of spherical mass M is given by: g = - GM/r^2. On or near the surface of the Earth, the gravitational field is uniform, so the value of g is approximately constant. Its value is equal to the acceleration of free fall.  The orbital period of a satellite is the time taken for one orbit. The orbital period can be found by equating the gravitational force GMm/r² to the centripetal force mv²/r. Kepler’s third law of planetary motion related the orbital period T to the orbital radius r: t²a r³. The orbital speed of a planet or satellite can be determined using the equation   v² = GM/r. Geostationary satellites have an orbital period of 24 hours and are used for telecommunications transmissions and for television broadcasting.

Define "gravitational field strength".

State Newton's Law of Gravitation.

Equate Newton's Law of Gravitation, defining all terms used.

Derive another equation used for gravitational field strength, defining all terms used.

Define the "orbital period" for a satellite.

How can the orbital period be found?

Name the two terms related by Kepler's Third Law.

Derive an equation for Kepler's Third Law.

How can the orbital speed of a satellite be determined?

Give some characteristics of geostationary satellites.

Many systems, mechanical and otherwise, will oscillate freely when disturbed from their equilibrium position.  Some oscillators have motion described as simple harmonic motion (abbreviated as s.h.m.) For these systems, graphs of displacement, velocity and acceleration against time are sinusoidal curves. (displacement = sin t, velocity = cos t, acceleration = -sin t respectively).  During a single cycle of s.h.m., the phase changes by 2π radians. The angular frequency ω of the motion is related to its period T and frequency f by the equations: ω = 2π/T and ω = 2πf. (f = 1/T). In s.h.m, displacement x can be represented as a function of time t by equations of the form: x = A sin (2πft)and x = A cos (2πft). A body executes simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position. The acceleration is always directed toward the equilibrium position. Acceleration ‘a’ is related to displacement ‘x’ by the equation a =- (2πf) ²x. The maximum speed v_max is given by the equation v_{max­ =} (2πf)A. The frequency or period of a simple harmonic oscillator is independent of its amplitude.  In s.h.m, there is a regular interchange between kinetic energy and potential energy. Resistive forces remove energy from an oscillating system. This is known as damping. Damping causes the amplitude to decay with time.  When an oscillating system is forced to vibrate close to its natural frequency, the amplitude of vibration increases rapidly. The amplitude is at a maximum when the forcing frequency matches the natural frequency of the system, this is resonance. Resonance can be a problem, but it can also be very useful.

Why will many systems oscillate freely?

Which type of motion do these systems undergo?

How is the angular frequency related to the period and the frequency?

How can displacement be represented as a function of time?

Define "simple harmonic motion".

What is the equation for simple harmonic motion?

What is the maximum speed equal to?

Which two types of energy are interchanged?

What is the process called where resistive forces try to remove energy?

Define "resonance" and state at least 2 uses and 2 problems with it.

The kinetic model of matter allows us to explain behaviour (e.g. changes of state) and macroscopic properties (e.g. specific heat capacity and specific latent heat) in terms of the behaviour of molecules. The kinetic theory of matterstates that all matter is made up of a large number of tiny atoms or molecules which in continuous motion. The internal energy of a system is the sum of the random distribution of kinetic and potential energies associated with the atoms or molecules that make up the system. If the temperature of an object increases, there is an increase in its internal energy. Internal energy also increases during a change of state, but there is no change in temperature. Temperatures on the thermodynamic (Kelvin) and Celsius scales of temperature are related by:  T (K) = θ (^oC) + 273.15, θ(^oC) = T (K) – 273.15. At absolute zero, all substances have a minimum internal energy. The word equation for the specific heat capacity of a substance is specific heat capacity = energy supplied/mass x temperature change. The specific heat capacity of a substance is the energy required per unit mass of the substance to raise the temperature by 1K (or 1^oC). The energy transferred in raising the temperature of a substance is given by         E = mcΔθ. The specific latent heat of a substance is the energy required per kilogram of the substance to change its state without any change in temperature.

What does the kinetic model allow us to do?

Define the "internal energy" of a system.

What increases the internal energy of a substance?

How are the Kelvin and Celsius scales related?

State the substance characteristics at absolute zero.

Define the "specific heat capacity" of a substance. State the equation used.

Define the "specific latent heat" of a substance.

Boyle’s Law: The pressure exerted by a fixed mass of gas is inversely proportional to its volume, provided the temperature of the gas remains constant. For an ideal gas pV/T = constant. One mole of any substance contains N_A particles (atoms or molecules). N_A = Avogadro constant = 6.02 x 10²³ mol^-1 The equation of state for an ideal gas is pV = NkT (for N atoms) or pV = nRT (for n moles). The mean translational kinetic energy E of a particle (atom or molecule) of an ideal gas is proportional to thermodynamic temperature T. The mean translational kinetic energy E is related to temperature T by the equation E = 3/2 kT, where k is the Boltzmann constant (1.38 x 10^-23J K^-1).

Define "Boyle's Law". State the equation used for Boyle's Law.

Define "Charles' Law". State the equation used for Charles' Law.

Define the pressure law. State the equation used in the pressure law.

What equation is derived when these are combined?

Define the mole.

What does one mole of any substance contain?

State the equation(s) of state for an ideal gas, defining any terms used.

How is the kinetic energy related to temperature? Define any terms used.