Further Mechanics
Radian Measures
☆ When dealing with circular motion, it is often easier to make use
of angular quantities.
☆ These make use of an angle unit, known as the radian.
☆ The conversion between degrees and radians is:
2π radians = 360o
☆ The equation to find the angle in radians you have turned through
is:
𝑠
θ = 𝑟 Where ‘𝑠’ is arc length and ‘𝑟’ is the radius of the circle
☆ Angular speed is given by:
𝑣
ω = 𝑟 ω = 2π𝑓
Circular Motion
☆ From Newton’s first law, we know that for an object to change
velocity, a resultant force must act.
☆ In circular motion, since the direction of the object is
continually changing, the velocity must also be changing.
☆ Therefore a resultant centripetal force is required.
☆ This force points towards the centre of the object’s orbit and is
given by the equation:
2
𝑚𝑣
F = 𝑟
☆ An alternative form of this equation can be given, which makes
use of the radian and angular speeds.
2
F = 𝑚ω 𝑟
☆ It is important to consider what is contributing to the
centripetal force at each point in the cycle.
☆ For a ball being spun on a string:
☆ At position A, the weight of the ball is directly
contributing to the centripetal force
since it is acting directly towards
the centre of the circle - this means
that the inwards force provided by the
string is at a minimum,
☆ At position B, the weight of the ball
is acting perpendicular to the
direction of the centripetal force,
meaning it makes no contribution and
the string provides the full force,
, Further Mechanics
☆ At position C, the weight of the ball is acting opposite to
the direction of the centripetal force, meaning the inwards
force of the string must overcome the weight and provide
the required centripetal force - this means it is at a
maximum.
Simple Harmonic Motion
☆ Simple harmonic motion is a mechanical process that is
characterised by the following conditions:
☆ The object oscillates either side of an equilibrium
position,
☆ A restoring force always acts towards this equilibrium
position,
☆ The force is proportional to the object’s displacement,
☆ Consequently the object has an acceleration proportional to
its displacement.
☆ The defining equation for SHM is:
F = -k x Where ‘x’ is displacement and ‘k’ is a constant
Further Equations
☆ You should understand the following terms in the context of SHM:
☆ The frequency is the number of full cycles that occur each
second,
☆ A full cycle is the motion from maximum positive
displacement, to maximum negative displacement and then
back to the maximum positive displacement again,
☆ The time period is the length of time it takes to complete
a cycle.
☆ Like in circular motion, SHM make use of ⍵, the angular
frequency.
☆ You need to be able to apply the following equations when
analysing SHM scenarios:
2
a = -ω x
x = A cosωt
v = -Aω sinωt
2
a = -Aω cosωt
Radian Measures
☆ When dealing with circular motion, it is often easier to make use
of angular quantities.
☆ These make use of an angle unit, known as the radian.
☆ The conversion between degrees and radians is:
2π radians = 360o
☆ The equation to find the angle in radians you have turned through
is:
𝑠
θ = 𝑟 Where ‘𝑠’ is arc length and ‘𝑟’ is the radius of the circle
☆ Angular speed is given by:
𝑣
ω = 𝑟 ω = 2π𝑓
Circular Motion
☆ From Newton’s first law, we know that for an object to change
velocity, a resultant force must act.
☆ In circular motion, since the direction of the object is
continually changing, the velocity must also be changing.
☆ Therefore a resultant centripetal force is required.
☆ This force points towards the centre of the object’s orbit and is
given by the equation:
2
𝑚𝑣
F = 𝑟
☆ An alternative form of this equation can be given, which makes
use of the radian and angular speeds.
2
F = 𝑚ω 𝑟
☆ It is important to consider what is contributing to the
centripetal force at each point in the cycle.
☆ For a ball being spun on a string:
☆ At position A, the weight of the ball is directly
contributing to the centripetal force
since it is acting directly towards
the centre of the circle - this means
that the inwards force provided by the
string is at a minimum,
☆ At position B, the weight of the ball
is acting perpendicular to the
direction of the centripetal force,
meaning it makes no contribution and
the string provides the full force,
, Further Mechanics
☆ At position C, the weight of the ball is acting opposite to
the direction of the centripetal force, meaning the inwards
force of the string must overcome the weight and provide
the required centripetal force - this means it is at a
maximum.
Simple Harmonic Motion
☆ Simple harmonic motion is a mechanical process that is
characterised by the following conditions:
☆ The object oscillates either side of an equilibrium
position,
☆ A restoring force always acts towards this equilibrium
position,
☆ The force is proportional to the object’s displacement,
☆ Consequently the object has an acceleration proportional to
its displacement.
☆ The defining equation for SHM is:
F = -k x Where ‘x’ is displacement and ‘k’ is a constant
Further Equations
☆ You should understand the following terms in the context of SHM:
☆ The frequency is the number of full cycles that occur each
second,
☆ A full cycle is the motion from maximum positive
displacement, to maximum negative displacement and then
back to the maximum positive displacement again,
☆ The time period is the length of time it takes to complete
a cycle.
☆ Like in circular motion, SHM make use of ⍵, the angular
frequency.
☆ You need to be able to apply the following equations when
analysing SHM scenarios:
2
a = -ω x
x = A cosωt
v = -Aω sinωt
2
a = -Aω cosωt
