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Science > 3.4 Mechanics > Flashcards

3.4 Mechanics Flashcards

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1
Q

Law of conservation of momentum

A

When there is no external net force, the total momentum of a system in a given direction (two objects) before the collision is equal to the total momentum after the collision in the same direction.

  • Momentum lost by object 1 = Momentum gained by object 2
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2
Q

Elastic collisons

A

Elastic collisions are collisions in which both momentum and kinetic energy are conserved. Therefore, when there is an elastic collision, the total system kinetic energy (two objects) before the collision is equal to the total system kinetic energy after the collision.

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3
Q

Momentum (p)

A

p = mv

  • Momentum is a vector
  • Momentum has the same direction as the velocity
  • The unit for momentum is kgms^-1
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4
Q

Impulse (△p)

A

△p = F△t
A force acting for a certain time (contact time) produces a change in momentum (impulse)

  • Impulse is a vector
  • Impulse has the same direction as the force
  • The unit for impulse is N s
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5
Q

Law of conservation of energy

A

The law of conservation of energy states that the total energy of an isolated system remains constant (is conserved over time). Therefore, energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.

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6
Q

To convert from ms to s

A

ms / 1,000 = s

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7
Q

Centre of mass

A

The center of mass is the point where the total mass of different particles of a system is concentrated (act through)

For two masses
m1r1 = m2r2
Xcom = m1x1 + m2x2/(m1+m2)

  • When using the second method, make the point you choose, the point from which you want to obtain the coordinate. Then, write answer ‘from (name of point chosen)’.
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8
Q

Units for torque

A

Nm

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9
Q

Units for impulse

A

Ns

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10
Q

Collision and centre of mass

A

If two objects collide and their velocities changed due to the collision, according to the law of conservation of momentum, Pi = Pf.

Deriving the formula gives us, Vci = Vcf.

In elastic collisions and explosions, the velocity of the centre of mass in a given direction does not change. The law of conservation of momentum is still valid.

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11
Q

Circular motion

A

For an object moving in uniform circular motion around the perimeter of a circle with a constant speed, the velocity is frequently changing. This is because velocity is a vector, having a constant magnitude, but a changing direction. The direction of the velocity at any instant is at a tangent to the circle.

As a result of the changing velocity, the object will experience centripetal acceleration, towards the centre of the circle. In accordance with Newton’s second law, the object will also experience a net centripetal force, in order to cause its acceleration. The direction of the net force at any instant is in the same direction as the acceleration.

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12
Q

To convert from degrees to radians

A

Degrees x π/180 = radians

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13
Q

Banked roads

A

A road in a bend is banked from outside to stop a car from sliding when it is traveling at a high speed. If the bank angle is 0, the surface is flat and the support force is vertically upwards, which means that friction is the only force which keeps the vehicle in its path. This friction must be large enough to provide the centripetal force for the car to drive in a circle of radius r.

Inclined edges on a road provide an additional centripetal force, which is the horizontal component of the vehicle’s support force (R), which keeps the vehicle in its path and prevents a car from being “pushed out” of the circle.

  • The vertical forces are balanced as the car does not move vertically.
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14
Q

Law of gravitation

A

The attractive force (gravitational force) between two masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two masses.

Fg ∝ Mm and F∝1/r^2
Deriving the formula gives us, Fg = GMm/r^2
Which is inverse square law

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15
Q

Gravitational constant

A

6.67 x 10^-11

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16
Q

Gravitational field strength

A

Gravitational field strength (g) is the attractive force of the earth on a 1kg mass (Nkg^-1) that is on its surface. Therefore, g = 9.829878576

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17
Q

To convert from T to kg

A

T x 1000 = kg

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18
Q

What is required to keep a satellite in orbit

A

Centripetal force is required to keep a satellite in orbit, which is provided by the gravitational force on the satellite. Therefore, for a satellite, centripetal force = gravitational force. (No other forces act on the satellite).

  • Subsequently, the centripetal acceleration of the satellite is provided by the same acceleration of gravity (gravitational field strength).
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19
Q

Gravitational field lines

A
  • Field lines are directed radially inwards, towards the centre of mass
  • Field lines become more spread out as the distance from the Earth increases, indicating the diminishing strength of the field
  • The gravitational field strength at a point in a field is independent of the mass placed there (it is a property of the field). Therefore, two objects of different mass placed at the same point in the field will experience the same field strength, but different gravitational forces
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20
Q

Units for gravitational field strength

A

ms^-2 and NKg^-1

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21
Q

Angular velocity and radians

A

An object in circular motion has an angular velocity (ω), as well as a linear velocity (v).

  • Angular velocity is the change in the number of radians per second during the rotation.
    ω = △θ / △t
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22
Q

Units for angular velocity

A

rads^-1

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23
Q

Conical pendulum

A

A mass attached to a string swings around in a circle so that the string forms a cone-shaped path. The vertical component of the tension force in the string is equal to the weight of the mass, while the horizontal component of the tension force supplies, and is therefore equal to the centripetal force towards the center.

v = √rgTanθ
T = 2πr/v
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24
Q

Does the absence of atmosphere mean there is no gravity? Why do astronauts feel weightless?

A
  • The absence of atmosphere does not mean that there is no gravity, as gravity continues to act in the vacuum of space.
  • Individuals are only aware of their weight due to reaction forces on the ground. For astronauts in free fall, the earth curves away as fast as they fall to the earth, and therefore they will not land. This means they will feel “weightless”, as they will not experience any reaction forces on the ground.
  • In reality, one is never completely “weightless”, (even in deep space), as there is no place that experiences 0ms^-2 of gravity.
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25
Q

To convert from s to days

A

s/86,400 = days

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26
Q

Rotational motion

A
  • Different linear speeds (measured in m/s), depending on how far they are from the axis of rotation
  • Same rotational speeds (measured in rad/s), no matter where they are located
  • Both objects travel the same angular distance θ, though did not travel the same tangential distance

Linear velocity = d/t = 2πr/T
Angular velocity = θ/t = 2π/T

To convert from linear to angular velocity, use the equation v = ωr

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27
Q

Angular displacement

A

Angular displacement is the angle (θ) formed by the object at a given time (t)

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28
Q

Equation for the circumference of a circle

A

2πr

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29
Q

Equation for area of a part of a circle

A

θr

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30
Q

Angular velocity

A

Angular velocity is the angle (θ) formed by the object in one second

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31
Q

Angular acceleration

A

Angular acceleration is the rate of change of angular velocity

a = rα

32
Q

Units for angular acceleration

A

rads^-2

33
Q

Wheel rolling without slipping, in a straight line.

A

The forward displacement of the wheel (d) is equal to the linear displacement (d) of a point fixed on the rim.

34
Q

Equations for distance, velocity and acceleration (rotational motion)

A 
d = rθ
v = rω
a = rα
35
Q

Torque for a single object

A

Torque produces rotational kinetic energy.

  • An object rotates with a force (F), which is perpendicular to the distance (r)
    τ = Fr
  • An object rotates with a force (F) and an acceleration (α)
    τ = Iα
36
Q

Rotational inertia/moment of inertia of an object

A

Rotational inertia (I) is the torque required to create an angular acceleration of one rads^-1 (resistance to rotation). Rotational inertia is dependent on m and r, (larger mass, or larger distance = larger inertia).

I = mr^2 (For if different objects in a system rotate around a point, otherwise I = 1/2mr^2)
or
I = τ/a

  • If two objects have the same mass and shape, this means that their rotational inertia will remain the same
37
Q

Units for rotational inertia

A

kgm^2

38
Q

Rotational inertia/moment of inertia of different objects in a system

A
  • Different objects in a system, rotate around a point with different masses (m) and distances (r) from each mass to the centre of rotation (o).
    I = Σ miri^2 + … (i = 1,2,3…)
39
Q

Angular momentum

A

Angular momentum (L) is the rotational momentum of an object. Equal and opposite torques produce equal and opposite changes in angular momentum.

L = Iω
or
L = mvr

  • Angular momentum only comes from an object which is rotating.
Li = Lf
l1ωi = (l1 + l2)ωf
40
Q

Units for angular momentum

A

kg m^2s^-1

41
Q

Law of conservation of angular momentum

A

When there is no external net torque, the total angular momentum of a system in a given direction (two objects) before the collision is equal to the total momentum after the collision in the same direction.

  • Angular momentum lost by object 1 = Angular momentum gained by object 2
42
Q

Skater spinning with arms out vs. arms in

A

From L = mvr, we can see that as angular momentum is conserved, extended arms results in a larger radius, decreasing her rotational velocity. Whereas, bringing in arms results in a smaller radius, increasing her rotational velocity.

As there are no external torques acting, her angular momentum is conserved. From I = mr^2 and L = Iω, we can see that as angular momentum is conserved, extended arms results in a larger radius, increasing her rotational inertia and decreasing her rotational velocity. Whereas, bringing in arms results in a smaller radius, decreasing her rotational inertia and increasing her rotational velocity.

43
Q

Rotational kinetic energy

A

Rotational kinetic energy depends on the angular velocity and rotational inertia of the object.

Ek = 1/2lω^2
Work done = τθ

  • The rotational kinetic energy gained by the object is equal to the work done on rotating the object

τθ = 1/2lω^2

44
Q

Total energy of a system

A

The total energy of a system is equal to the sum of the translational and rotational kinetic energy

Et = 1/2mv^2 + 1/2lω^2

45
Q

Rotational inertia and rolling objects

A
  • Spheres of a different size or mass will accelerate equally when rolled down an incline, and therefore they will roll with the same velocity
  • A solid cylinder, hollow cylinder and a sphere will roll down a slope with different velocities
  • A cylinder with a smaller rotational inertia will roll more quickly than a cylinder with a larger rotational inertia, as it will require more time to start rolling. A hollow cylinder has a larger rotational inertia than a solid cylinder, this is because its mass is located farthest from the axis of rotation. Therefore, the hollow cylinder will take longer to reach the bottom when rolled down an incline.
46
Q

Restoring force

A

Restoring force is the force that aims to restore an oscillating object to its equilibrium (rest) position. The restoring force creates acceleration on the moving object.

47
Q

Mass oscillating vertically on a spring

A
  • The net force in the spring acts towards the centre (O), while the object is moving the opposite direction (a/F and y are always acting in opposite directions). Acceleration (a) also acts towards the centre (O).
48
Q

Simple harmonic motion

A

Motion is known as simple harmonic motion if the restoring force (F) on an object is directly proportional to its displacement (y), and is opposite to the direction of the motion of the object.

49
Q

Features of simple harmonic motion

A
  • Restoring force (F) and acceleration (a) always act towards the centre (O) of motion.
  • The further the object moves from the centre (y), the greater the restoring force (F) and the acceleration (a) of the object.
  • The restoring force (F) and the acceleration (a) are both directly proportional to the displacement (y).
  • The restoring force (F) and the acceleration (a) are at a maximum when the displacement (y) is at a maximum.
50
Q

Position time graph for simple harmonic motion

A
  • The amplitude of the curve is the maximum displacement
  • The equilibrium line is the x axis
  • The time period (T) is the time for one complete oscillation (one crest and trough)
51
Q

Position time graph for simple harmonic motion summary

A
  1. When t = 0, the object is at the centre and the displacement is Y = 0 (at its minimum)
  2. When t = T/4, θ = 90, the displacement is Y = A (at its maximum, at the top)
  3. When t = 3/4 T, θ = 270, the displacement is Y = -A (at its maximum, at the bottom)
  4. When t = T, θ = 360, t = 0, and so the cycle starts again and simple harmonic motion continues
52
Q

Sign conventions

A
  • All measurements (y, v and a) are measured from the centre of the motion. Values taking the direction upward from the centre are positive, and downward from the centre are negative.
53
Q

Phasors

A

It is assumed that phasors are rotating in an anticlockwise direction. For the displacement phasor, the length of each phasor arm is directly related to the amplitude of the wave, and the angle between the phasors is the same as the angle of phase difference between the sine waves.

54
Q

Simple Harmonic Motion - Velocity

A

1 - Velocity is at maximum
2 - Velocity is 0
3 - Velocity is at maximum and negative
4 - Velocity is 0

55
Q

Simple Harmonic Motion - Acceleration

A

1 - Acceleration is 0
2 - Acceleration is at maximum and negative
3 - Acceleration is 0
4 - Acceleration is at maximum

56
Q

When is acceleration negative?

A

At the top end, the acceleration is negative because the restoring force acts towards the centre (downwards), and thus, acceleration also acts downwards. As vectors measuring upwards are taken as positive, acceleration is negative.

57
Q

Application of SHM

1. Mass on a spring

A

T = 2π√m/k = 1/f

F = -kx = ma
(negative signs indicate that k and x act in opposite directions)
a = -kx/m = -w^2x

Notes
- If there are two springs, the same formula may be used, though first add the spring constants of both springs together to calculate the total value of k

58
Q

Application of SHM

2. Simple pendulum

A

T = 2π√L/g = 1/f

F = mg sinθ

a = -w^2y

Notes
- The motion is only considered to be simple harmonic if the angle is very small (small angle approximation)

59
Q

Application of SHM

3. Liquid in a u-tube

A

T = 2π√L/g = 1/f

F = -2Aypg = ma

a = -(2Apg)y = -w^2x

60
Q

Application of SHM

4. Torsional pendulum

A

T = 2π√I/k = 1/f

α = -w^2θ

61
Q

Energy in simple harmonic motion

A

An object with SHM has both kinetic energy and potential energy.

Total energy at any point
1. For a spring
1/2mv^2 + 1/2kx^2
- There is maximum potential energy when the spring is compressed/stretched fully

  1. For a pendulum
    1/2mv^2 + mgh
    - There is maximum potential energy when the pendulum is at each peak
62
Q

Damped SHM

A

Damped SHM occurs when due to air resistance and friction, the amplitude of the SHM decreases with time, (though the period stays the same). As a result of this, the displacement/time (y/t) graph of the SHM takes the shape of a decay envelop.

63
Q

Forced SHM

A

Forced SHM occurs when due to an external driving force subjected on an object, SHM is triggered.

64
Q

Resonance

A

Resonance is the stage when the natural frequency of an object is equal to the driving frequency. The frequency at resonance is the resonant frequency. The amplitude of the oscillation at resonance is the maximum.

65
Q

Applications of resonance

A
  1. Musical instruments (strings & woodwinds) produce sounds by resonant vibrations
  2. Radios/TV can be tuned to correct channels by having a resonant current and a frequency
  3. Buildings and bridges may vibrate in resonant frequencies, giving maximum amplitudes (must be designed so that they do not reach resonant frequency during natural disasters).
66
Q

Maximum acceleration in simple harmonic motion

A 
a = Aω^2
ω = 2π/T

or

a = -ky

Acceleration is directly proportional to displacement from the equilibrium position.

67
Q

Maximum velocity in simple harmonic motion

A

v = Aω

68
Q

Maximum displacement in simple harmonic motion

A

y = A

69
Q

How to calculate the time taken to travel from a point to a position of maximum displacement

A
  • Draw point (t) on reference circle and draw triangle with measurement of amplitude and distance from equilibrium position to position of maximum displacement
  • Calculate the angle between the phasor and central line in radians (set calculator)
  • Use θ = ωt, to calculate t
70
Q

How to increase the amplitude of a pendulum

A

A greater driving force must be produced on the pendulum. If the direction of this driving force is the same as the restoring force, and the frequency of the driving force will match the natural frequency of the pendulum. Because of this, the energy put into the system will build up and produce a larger amplitude in the pendulum.

71
Q

What is the period (T) of a pendulum determined by?

A

The period (T) of a pendulum determined by gravity.

72
Q

Banked road vector triangle

A

←Fc = mv^2/r ↓W = mg

↖R

73
Q

How to adjust a mass on a spring to keep the same large amplitude when the mass is decreased

A

The machine is resonating at (_Hz) - this is when the natural frequency of the system matches that of the external oscillator. If this system is a mass on a spring it has a natural frequency of T = 2π√m/k. If the mass is changed, the time period (and hence the natural frequency) will change also. The time period should be (_s). If the mass (m) is decreased, the spring constant (k) should be decreased in proportion to 80% to keep m/k constant, and hence the natural frequency constant.

74
Q

Calculate the time from a position of maximum displacement to a given distance (of an oscillating spring)

A
  1. Draw reference circle
  2. Calculate the angle (θ) and convert the answer to radians (x π/180)
  3. Use ω = 2πf, where the frequency is the rate of oscillation in Hz
  4. Use t = θ/ω
75
Q

How can two different objects with different masses in the same orbit have the same velocity?

A

The velocity of the objects only depends on the mass of the earth and their radius of orbit, not their individual masses. This because…

Fc = Fg
mv^2/r = GMm/r^2 (m and r^2 cancel out)
v^2 = GM/r

Therefore, because both objects follow the same orbit, they both have the same radius of orbit and as the mass of the earth remains constant, they will have the same velocity.

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