Further mechanics Flashcards
What do we mean by uniform circular motion?
motion in a circle at a constant speed
What is a requirement for circular motion to occur?
the force must constantly act perpendicular to the objects velocity (towards the centre of the circle)
What is the speed called for an object in uniform circular motion?
linear speed
What is something to know about the velocity of an object in circular motion?
always acts at a tangent to the objects’ path
What do we mean if the acceleration is centripetal? What are some things to know about the direction of the acceleration?
-acts towards the centre of the circular path
-the acceleration is perpendicular to the direction of the linear speed
Is the velocity constant during uniform circular motion?
no, velocity is always changing due to changes in direction (as its a vector quantity)
What do we mean if something is centripetal?
it acts towards the centre of the circular path
What do we mean when a force is centripetal?
a force that acts towards the centre of the circular path
Draw a diagram of a particle undergoing uniform circular motion with the centripetal force, acceleration and velocity acting on it
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/09/12.2.1.1-Force-and-acceleration-direction.png
What do we mean by the angular displacement (θ) of a body in circular motion? What is it measured in?
-the change in angle of a body as it rotates around a circle
-measured in radians
How can we work out the angular displacement of an object? (2 equations)
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.1.2-Radians-Equation-1.png
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.1.2-Radians-Equation-2.png
where:
Δθ = angular displacement, or angle of rotation (radians)
s = length of the arc, or the distance travelled around the circle (m)
r = radius of the circle (m)
-both distances must be in metres
What is a radian (rad)?
the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle
What is a relationship between an angle, a radius and the arc length?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/09/12.1.1.1-Radians-definition.png
-when the angle is equal to one radian, the length of the arc (Δs) is equal to the radius (r) of the circle
How can we work out arc length from an object moving in a circular motion?
Δs=r x Δθ
where:
Δs= arc length (m)
r= radius (m)
Δθ= angular displacement (rad)
How can we convert degrees into radians? What about vice cersa?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.1.2-Radians-Equations-5.png
-rearrange equation to find angle in degrees
What do we mean by angular speed? What is its’ symbol
(⍵- omega)
the rate of change in angular displacement with respect to time
Is angular speed a vector or scalar quantity? What is it measured in?
-scalar
-rad s^-1
How can we calculate angular speed? (2)
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.1.3-Angular-Speed-Equation-2.png
where:
Δθ = change in angular displacement (radians)
Δt = time interval (s)
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.1.3-Angular-Speed-Equation-1.png
where:
v = linear speed (m s^-1)
r = radius of orbit (m)
T = the time period (s)
f = frequency (Hz)
Draw a diagram showing a particle in uniform circular motion with its angular speed and linear speed
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/10/12.1.1.2-Angular-speed-diagram_1.png
What is centripetal acceleration defined as?
the acceleration of an object towards the centre of a circle when an object is in motion (rotating) around a circle at a constant speed
How can we calculate centripetal acceleration? (3)
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/09/3.-Calculating-Centripetal-Acceleration-equation-1-1.png
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/09/3.-Calculating-Centripetal-Acceleration-equation-2.png
-https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/09/3.-Calculating-Centripetal-Acceleration-equation-4.png
where:
a = centripetal acceleration (m s^−2)
v = linear speed (m s^−1)
⍵ = angular speed (rad s^−1)
r = radius of the orbit (m)
How can we calculate linear speed?
v = r⍵
where:
v= linear speed (ms^-1)
r= radius (m)
⍵= angular speed (rad s^-1)
What is the centripetal force defined as?
the resultant force towards the centre of the circle required to keep a body in uniform circular motion
-always directed towards the centre of the body’s rotation.
What would happen if to an object in circular motion if the centripetal force didn’t exist anymore?
the object will move in the direction of the velocity at that point and fall in the very same direction no longer moving in a circular path
How can we calculate the centripetal force?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2020/10/4.-Calculating-Centripetal-Force-equation-1.png
where:
F = centripetal force (N)
v = linear velocity (m s^-1)
⍵ = angular speed (rad s^-1)
r = radius of the orbit (m)
Give examples of centripetal forces in these scenarios:
-car travelling around a roundabout
-ball attached to a rope moving in a circle
-earth orbiting the sun
-friction between car tyres and the road
-tension in the rope
-gravitational force
For a car rotating and being able to hold a corner, what is needed?
the limiting frictional (centripetal) force must be enough to maintain the motion
How can we work out the limiting frictional (centripetal) force which prevents an object in circular motion (on a flat surface) from slipping?
F= µmg
where
f= limiting frictional force
µ= coefficient of friction
mg= weight (on a flat surface)
How can we work out the maximum speed of an object (on a flat surface) in circular motion before it slips?
v= õgr
where:
v= max speed of object (ms^-1)
µ= coefficient of friction
g= gravitational field strength (Nkg^-1)
r= radius (m)
Why do banked paths increase the speed of an object compared to flat paths?
-flat paths only consist of one centripetal force
-When a car/airplane moves on a banked pathway, the normal reaction force is at an angle so the horizontal component provides some centripetal force meaning less is provided from friction
-this means the car/airplane can move faster around the bend before the max frictional force is reached
How can we find the speed of an object moving on a banked pathway? Derive the equation explaining each step
-first split the normal force into its components
-find tanθ by doing sin/cos with the components (1)
-as the weight is equal to the vertical component, substitute the vertical with mg
-rearrange to find the horizontal component (2)
-as the mgtanθ is equal to the horizontal component, it is also equal to the centripetal force equation which can be rearranged to find the speed (3, 4)
1) tanθ= normal horizontal/ normal vertical
2) mgtanθ- normal horizontal force
3) mgtanθ= mv^2/r
4) v= √grtanθ (as the mass cancels out on both sides)
What is simple harmonic motion
a specific type of oscillation
What do we mean by equilibrium in SHM?
the position of an object where no resultant force is acting on it
What do we mean by a restoring force?
a force which acts to bring a body back from equilibrium
What are the two conditions required for SHM to occur?
-acceleration of the oscillating object is proportional to the displacement
-the acceleration of the oscillating object must be in opposite directions to the displacement
Give some examples of oscillators which undergo SHM (4)
-the pendulum of a clock
-a mass on a spring
-guitar strings
-the electrons in alternating current flowing through a wire
What equation can be used to show the relationship between acceleration and displacement in SHM?
a ∝ −x
What is something to know about the direction of the restoring force in SHM?
-always moves in opposite direction to displacement
-the restoring force and acceleration always act in the same direction (towards equlibrium)
Draw and label a diagram of an oscillating pendulum consisting of the restoring force, acceleration and displacement
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/17.1-SHM-pendulum.png
How can we find the acceleration of an object oscillating in SHM from a graph?
a= -A ⍵^2 cos (⍵t)
where:
a= acceleration (ms^-2)
A= amplitude (m)
⍵= angular frequency (rad s^-1)
t= time (s)
How can we find the acceleration of an object oscillating in SHM?
a = −⍵^2 x
where:
a = acceleration (m s^-2)
⍵ = angular frequency (rad s^-1)
x = displacement (m)
How can we find the acceleration of an object oscillating in SHM when the object is at max displacement?
substitute the displacement with the amplitude, don’t multiply equation with the -1
How can we find the displacement of an object oscillating in SHM from a graph (from its maximum displacement)?
x= A cos (⍵t)
where:
x= displacement (m)
A= amplitude (m)
⍵ = angular frequency (rad s^-1)
t= time (s)
How can we find the displacement of an object oscillating in SHM from a graph (from its equilibrium position)?
x = A sin (⍵t)
where:
x= displacement (m)
A= amplitude (m)
⍵ = angular frequency (rad s^-1)
t= time (s)
How can we find the max displacement for an object in SHM?
x max= amplitude
How can we find the velocity of an object oscillating in SHM from a graph?
v= -A ⍵ sin (⍵t)
where:
where:
v= velocity (ms^-2)
A= amplitude (m)
⍵ = angular frequency (rad s^-1)
t= time (s)
How can we find the velocity of an object oscillating in SHM?
v= ±⍵√(A^2-x^2)
where:
v = speed (m s^-1)
A = amplitude (m)
⍵ = angular frequency (rad s^-1)
x = displacement (m)
How can we find the maximum velocity of an object oscillating in SHM?
v max= A⍵
where:
v = speed (m s^-1)
A = amplitude (m)
⍵ = angular frequency (rad s^-1)
When is the velocity of an oscillating object in SHM at the maximum?
at the equilibrium position (when x=0)
What can a mass-spring system be used for in SHM?
can be used as a simple harmonic oscillator
How can we work out the time period of a mass-spring system in SHM?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.2.4-Mass-Spring-Time-Period-Equation.png
where:
T = time period (s)
m = mass on the end of the spring (kg)
k = spring constant (N m^-1)
What can a simple pendulum be used for in SHM? What is it made of?
-another type of simple harmonic oscillator
-the pendulum consists of a string and a bob (a weight, generally spherical) at the end
What is something needed to know when oscillating a simple pendulum? Why
-the angle by which the pendulum is displaced must be less than 10°
-this is due to the derivation of the time period formula where a small angle approximation is used, so for larger angles this approximation is no longer valid, therefore isn’t a good model
How can we work out the time period of a simple pendulum?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.2.5-Period-of-Pendulum-Equation-_2.png
where:
T = time period (s)
L = length of string (m)
g = gravitational field strength (N kg^-1)
What types of energy are exchanged during SHM?
-kinetic
-potential (could be in many forms e.g. elastic/gravitational energy)
Draw an energy-displacement graph of the kinetic and potential energy during 1/2 a cycle of SHM. Explain at which points the energies are at a maximum
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/05/6.2.7-Energy-graph-with-displacement.png
-the kinetic energy is at a maximum when the displacement x = 0 (equilibrium position)
-the potential energy is at a maximum when the displacement is at a maximum x = A (amplitude)
What is something to know about the total energy during SHM?
the total energy of a simple harmonic system always remains constant and is equal to the sum of the kinetic and potential energies
Draw an energy-time graph of the energies during SHM
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/17.1-Energy-graph.png
SHM
RP7: (mass-spring system) Describe this practical
-calculate the spring constant of a mass-spring system
-done by investigating how the time period of the oscillations varies with the mass
SHM
RP7: (mass-spring system) What is the y=mx equation?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.2.8-Spring-Constant-from-Graph.png
SHM
RP7: (simple pendulum) Describe this practical
-calculate the acceleration due to gravity from a simple pendulum
-done by investigating how the time period of its oscillations varies with its length
SHM
RP7: (simple pendulum) What is the y=mx equation?
https://cdn.savemyexams.co.uk/cdn-cgi/image/w=1920,f=auto/uploads/2021/04/6.2.8-g-from-Graph.png
What is damping?
where the energy in an oscillator is lost to the environment, leading to reduced amplitude of oscillations
What are the three types of damping?
-light damping
-critical damping
-heavy damping
What do we mean by light damping?
-where the amplitude gradually decreases by a small amount each oscillation
-also known as under-damping
What do we mean by critical damping?
reduces the amplitude to zero in the shortest possible time (without oscillating)
What do we mean by heavy damping?
-where the amplitude reduces slower than with critical damping, also without oscillating
-also known as over-damping
Why does damping occur?
due to resistive forces, such as friction or air resistance, which act in the opposite direction to the motion, or velocity, of an oscillator
What is a key feature of SHM regarding damping?
frequency of damped oscillations don’t change
How do free oscillations occur?
-when there is no transfer of energy to or from the surroundings
-happens when an oscillating system is displaced and then left to oscillate
What is a free oscillation?
an oscillation where there are only internal forces acting and there is no energy input
Why does damping occur?
due to resistive forces, such as friction or air resistance, which act in the opposite direction to the motion, or velocity, of an oscillator
What is a key feature of SHM regarding damping?
frequency of damped oscillations don’t change
What are forced oscillations?
oscillations acted on by a periodic external force where energy is given in order to sustain oscillations
What is a free oscillation?
an oscillation where there are only internal forces acting and there is no energy input
What do we mean by the driving frequency?
the frequency of forced oscillations
What do we mean by the natural frequency?
the frequency of an oscillation when the oscillating system is allowed to oscillate freely
What is resonance?
where the amplitude of oscillations of a system drastically increase due to gaining an increased amount of energy from the driving force
How does resonance occur?
when the driving frequency is equal to the natural frequency of a system
What are some applications of resonance? (3)
instruments- an instrument such as a flute has a long tube in which air resonates, causing a stationary sound wave to be formed
radio- these are tuned so that their electric circuit resonates at the same frequency as the desired broadcast frequency
swing- if someone pushes you on a swing they are providing a driving frequency, which can cause resonance if it’s equal to the resonant frequency and cause you to swing higher
How does damping affect resonance?
-reduces the amplitude of resonance vibrations
-Note: the natural frequency f0 of the oscillator will remain the same
What are some negative effects of resonance? Give two examples
-can cause damage to a structure
-e.g. on a bridge, when people cross, the driving frequency will be close to the natural frequency meaning the bridge would oscillate violently
-this could be very dangerous and damage the bridge
-e.g. glass tube smashing from a sound wave at a driving frequency close to the natural frequency causing large oscillations
