Circular Motion (Chap.7) :: Physics (A-Level)
Circular Motion (Chap.7) :: Physics (A-Level)
Circular Motion (Chap.7) :: Physics (A-Level)
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Physics (A-level)
Both the centripetal acceleration and force are towards the center
(90 to that of the instantaneous velocity)
Figure 7.4 & 7.5 shows the angle between two radii OA and OB & vA
and vB (∆ )
Triangles OAB and CDE are similar
Consider angle ∆ to be so small that arc AB approximated as
a straight line
DE/CD = AB/OA
∆v/vA = ∆s/r
∆v = ∆s(vA/r) and dividing both sides by ∆t
∆v/∆t = (∆s/∆t)(vA/r)
A = v2/r
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The right hand-side of the equation shows the constants (π and G), where M is the
same (mass of the sun in the e.g.) when we are considering the relation between T
and r
Kepler’s third law of planetary motion states that for planet or satellites describing
circular orbits about the same central body, the square of the period is proportional to the
cube of the radius of the orbit (T2 r3)
Geostationary orbit refers to communication satellites (called geostationary satellites) that
are in equatorial orbits with exactly the same period of rotation as the Earth (24 hours),
and move in the same direction as the Earth (west to east) so that they are always above
the same point on the Equator
The gravitational field strength at a point is defined as the force per unit mass acting on a
small mass placed at that point
Newton’s second law: F = ma. Thus the gravitational field strength is given by g = F/m
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For small distances above the Earth’s surface, g is approximately constant and is called the
acceleration of free fall
Gravitational potential at a point in a gravitational field is defined as the work done per
unit mass in bringing a unit mass from infinity to the point
Example questions:
A satellite is orbiting the Earth. For an astronaut in the satellite, his sensation of weight is caused by
the contact force from his surroundings. The astronaut reports that he is ‘weightless’, despite being in
the Earth’s gravitational field. Suggest what is meant by the astronaut reporting that he is ‘weightless’.
Explain why the centripetal force acting on both stars has the same magnitude.
Oscillations (chap.13):
The time taken for one complete oscillation or vibration is referred to as the period T of
the oscillation
The number of oscillations or vibrations per unit time is the frequency f
Frequency f = 1/T
The distance from the equilibrium position is known as the displacement (vector quantity)
The amplitude (scalar quantity) is the maximum displacement
Simple harmonic motion is defined as the motion of a particle about a fixed point such
that its acceleration is proportional to its displacement from the fixed point, and is directed
towards the point
X0 → amplitude of oscillation
V → the gradient of displacement-time graph
leading to:
hence:
A particle is said to be undergoing free oscillations when the only external force acting on
it is the restoring force (vibrating at its natural frequency):
No force to dissipate energy, hence constant amplitude and total energy remains
constant, so s.h.m. are free oscillations
In real situations, however, resistive forces cause the oscillator’s energy to be dissipated,
eventually converted into thermal energy. The oscillations are said to be damped
Light damping: the amplitude decreases gradually with time (T of the oscillation is
slightly greater than the corresponding free oscillation)
Heavy damping: the oscillations will die away more quickly
Critically damped: the displacement decreases to zero in the shortest time, without
any oscillation
Overdamping: the displacement decreases to zero in a longer time than for critical
damping
A solid has fixed volume and shape (particles are close together, tightly bonded to their
neighbors, and vibrating about fixed points)
During transition between solid and liquid, the energy supplied does not increase
the K.E, hence the temperature of the solid, instead it is used to overcome the
intermolecular forces between the atoms or molecules – increasing the electrical
potential energy of the molecules, this increase is the latent heat of fusion of solid
A liquid has fixed volume, no fixed shape and similar density as to solid
During transition between liquid and gas, the intermolecular forces in the liquid
must be overcome, the latent heat of vaporization
The graph shows that:
The electrical potential energy of two
atoms very close together is large and
negative
As the separation increases, their
potential energy also increases
When atoms are completely separated,
their potential energy is maximum and
has a value of zero
A gas has no fixed shape or volume (widely
separated and free to move around within their
container)
Latent heat of vaporization > latent heat of fusion, due to the greater energy required to
completely separate the molecules than to break the rigid bonds in the solid (melting
involves breaking of fewer bonds per molecule); energy is required to push back the
atmosphere as liquid turns to vapour, vol. of vapour > vol. of liquid
During evaporation, the most energetic molecules are most likely escape the surfaces of
the liquid and hence reducing the average K.E. thus its temperature (cooling effect)
The internal energy of a system is the sum of the random distribution of kinetic and
potential energies of its atoms or molecules
Can be increased by heating and/or compression
First law of thermodynamics: The increase in internal energy of a body is equal to the
thermal energy transferred to it by heating plus the mechanical work done to it
∆U = q + w
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Specific heat capacity: The energy required per unit mass of a substance to raise its
temperature by 1 K (or 1 °C). Unit: J kg−1 K−1
Second reading:
Subtracting:
Specific latent heat of fusion: The energy required per unit mass of a substance to change
it from solid to liquid without a change in temperature. Unit: J kg−1
Specific latent heat of vaporization: The energy required per unit mass of a substance to
change it from liquid to gas without a change in temperature. Unit: J kg−1
Second reading:
Subtracting:
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Figure 17.2 shows for point charges Q1 and Q2 a distance r apart. Coulomb’s law gives the
force F as:
When the charges are in a vacuum (or air), the quantity is called the permittivity of
free space
Electric field strength: The force per unit positive charge at a point. Unit: Vm−1 or NC−1
The electric field E at the location of q is given by E = F/q, hence the electric field due to the
isolated point charge is:
Electric potential: The energy per unit charge due to a charged body’s position in an
electric field. Unit: V (volt)
Similar to electric field strength, electric potential is defined as the potential energy per
unit positive charge (e.g. at point A, having potential energy EPA):
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The electric field strength is equal to the negative of the potential gradient at that point:
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
P1V1 = P2V2
Charles’s law: The volume occupied by a gas at constant pressure is directly proportional
to its thermodynamic (absolute) temperature
Another series of experiments provide (using the number n of moles of the gas):
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or
Avogadro constant:
Amount of substance containing NA particles/molecules/atoms
Amount of substance which contains the same number of
particles/molecules/atoms as there are atoms in 12g of carbon-12
R, molar gas constant or universal gas constant (same value for all gases), having
the value of
An ideal gas is one which obeys the equation of state pV = nRT or pV = NkT
Mole: The amount of matter which contains the same number of atoms/nuclei as there are
in 12 g of carbon-12
Tiny pollen grains suspended in water shows the jerky, erratic, random motion (Brownian
motion), due to the bombardment from all sides of the water molecules. Brownian motion
can be reproduced by observing the motion of tiny soot particles in smoke, these particles
move in a jerky motion too, proving the idea of rapid, random motion as required by the
molecular model
The kinetic theory of gases is a theory which links these microscopic properties of particles
(atoms or molecules) to the macroscopic properties of a gas
The assumptions of the kinetic theory of an ideal gas are:
Time of collisions negligible compared to time between collisions
No intermolecular forces except during collisions
Random motion of molecules
Large number of molecules
Total volume of molecules negligible compared to volume of containing vessel /
Average separation large compared with the size of the molecules
Molecule in a box:
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Consider a collision in which a single molecule with mass m moving with speed c
parallel to one side of the box
Collision striking side ABCD of the cube; elastically rebounded to the opposite
direction (velocity is –c; momentum changes from mc to –mc), so momentum from
the single collision is:
Between consecutive collisions with side ABCD, the molecule travels a distance of 2l
at speed c. Hence:
Using Newton’s second law:
For the large number N of molecules, we write the average value of c2 as < c2>, and
multiply by N to find the total pressure:
So pressure of gas depends only on its density and the mean square speed of its
molecules
The mean translational kinetic energy of an atom (or molecule) of an ideal gas is
proportional to the thermodynamic temperature:
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The quantity is called the root-mean-square (r.m.s.) speed (crms) of the molecules
Since are constants, the average value of P will depend on the average value of
, which is 1/2, hence the average power <P> delivered to the resistor is:
The alternating current in the primary coil gives rise to an in phase alternating magnetic
flux, which threads through the secondary coil, in turn causes an induced e.m.f in the
secondary coil – Faraday’s law of electromagnetic induction
For an ideal transformer (100% efficient):
Full-wave rectification involves four diodes, and is referred to as a bridge rectifier circuit
The inputs terminals are P and Q. If P is positive during the first half-cycle, diodes 1 and 2
will conduct; in the next half-cycle, Q is positive so diodes 3 and 4 will conduct. Thus the
resistor will always have its upper terminal positive and its lower terminal negative.
However, the output is not good a good approximation to the steady voltage, hence a
capacitor across the output terminals is used:
The capacitor charges up on the rising part of the half-cycle, then discharges through the
resistor as the output voltage falls; is done to reduce the fluctuations in the unidirectional
output, where the process is called smoothing
The important factor is the time constant of the resistor-capacitor circuit. If the product of
the capacitance C and the load R is much larger than the half-period of the original supply,
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the ripple on the direct current or voltage will be small. Reducing the time constant will
increase the ripple: