Dynamics Flashcards
(186 cards)
1
Q
What are Cartesian coordinates?
A
A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, usually denoted as (x, y).
2
Q
What are polar coordinates?
A
A coordinate system where each point is determined by a distance from a reference point and an angle from a reference direction.
3
Q
How do you derive displacement, velocity, and acceleration quantities for circular motion?
A
By using angular velocity and angular acceleration formulas.
4
Q
What is angular velocity?
A
The rate of change of angular displacement, typically measured in radians per second (rad/s).
5
Q
What is angular acceleration?
A
The rate of change of angular velocity, typically measured in radians per second squared (rad/s²).
6
Q
What is the formula for linear motion with constant acceleration?
A
s = ut + ½ at².
7
Q
What indicates the positive horizontal and vertical components in Cartesian coordinates?
A
i and j, respectively.
8
Q
What are the radial and tangential directions in polar coordinates?
A
- Radial direction (from O to P) * Tangential direction (perpendicular to radial component)
9
Q
What is the unit vector in the radial direction in polar coordinates?
A
eᵣ.
10
Q
What is the unit vector in the tangential direction in polar coordinates?
A
eₜ.
11
Q
What is tangential velocity in circular motion?
A
The component of velocity that is tangent to the circular path.
12
Q
What is centripetal acceleration?
A
The acceleration directed towards the center of the circular path that keeps the particle moving in a circle.
13
Q
Fill in the blank: The standard results for circular motion with constant angular acceleration are derived from _______.
A
[angular acceleration].
14
Q
True or False: In circular motion, the radial acceleration is always zero.
A
False.
15
Q
What does the symbol α represent in circular motion?
A
Angular acceleration.
16
Q
What does the symbol ω represent in circular motion?
A
Angular velocity.
17
Q
How is the position of a particle in polar coordinates represented?
A
By its distance R from the origin O along with the angle θ.
18
Q
What is the significance of the negative sign in angular motion calculations?
A
It indicates a change in direction or a decrease in angular displacement.
19
Q
What does the term ‘tangential acceleration’ refer to?
A
The acceleration that is tangent to the circular path of the particle.
20
Q
What is the relationship between radial velocity and radius in circular motion?
A
Radial velocity is non-zero if the radius varies.
21
Q
What is the importance of rotational motion?
A
Rotational motion is crucial for understanding the dynamics of objects in motion and their interactions.
22
Q
What is the formula for tangential velocity in circular motion?
A
Rω
23
Q
What is angular velocity measured in?
A
radians per second (rad/s)
24
Q
What is the formula for radial acceleration in circular motion?
A
Rω²
25
What does the moment (or torque) represent in rotational mechanics?
The rotational equivalent of forces.
26
How is the moment of force calculated?
Moment = Force x perpendicular distance
27
What is the relationship between moment and force?
M = rF
28
What is Newton's 2nd Law in rotation?
F = Mα
29
What does the moment of inertia depend on?
Mass and its distribution about the axis considered.
30
What is the parallel axis theorem used for?
To calculate the moment of inertia about an axis parallel to the axis through the center of mass.
31
What is the formula for the perpendicular axis rule?
Jz = Jx + Jy
32
What is the significance of the moment of inertia?
It determines how much moment is required to accelerate a system.
33
Fill in the blank: The moment of inertia of a body is denoted by _______.
J
34
True or False: An object with a larger moment of inertia requires more torque to achieve the same angular acceleration.
True
35
What does the resultant moment of inertia represent?
The combined moment of inertia of all parts of a complex body.
36
What is the equation of motion according to Newton's 2nd law?
F = ma
37
What is the first law of Newton?
A particle will maintain its state of rest or uniform motion unless acted upon by a force.
38
What is the second law of Newton?
The acceleration of a particle is proportional to the resultant applied force.
39
What is the third law of Newton?
For every action, there is an equal and opposite reaction.
40
What is a free body diagram (FBD)?
A graphical representation showing all forces acting on an object.
41
What is static equilibrium?
The condition where the sum of all forces acting on an object is zero.
42
What happens to tension in a wire when supports are placed further apart?
Tension increases.
43
Fill in the blank: In contact forces, the two forces acting when two surfaces are in contact are the normal force and _______.
friction force
44
What is the formula for calculating friction force?
F_friction = μN
45
What are the components of a free body diagram?
Weight, normal force, friction force, applied force.
46
What is required for a system to be in static equilibrium?
The sum of forces in any direction must equal zero.
47
What is the role of the moment of inertia in rotational dynamics?
It quantifies the resistance of an object to changes in its rotational motion.
48
What is the result of applying Newton's 2nd law in rotational dynamics?
It allows calculation of angular acceleration and torque.
49
What is the equation of motion for mass m1?
T1 - m1g = m1a
## Footnote
This equation describes the forces acting on mass m1, where T1 is the tension, m1 is the mass, g is the acceleration due to gravity, and a is the acceleration.
50
What is the equation of motion for mass m2?
m2g - T2 = m2a
## Footnote
This equation describes the forces acting on mass m2, where T2 is the tension acting on m2.
51
What happens when the rotational inertia is negligible?
T2r - T1r = 0
## Footnote
This leads to the conclusion that T1 = T2 when rotational inertia (J) is considered negligible.
52
If T1 = T2, what is the relationship between T and the masses?
T = m1a + m1g
## Footnote
This equation allows for the calculation of the tension T in terms of the acceleration and weight of mass m1.
53
What is the formula for acceleration (a) derived from the equations of motion?
a = (m2 - m1)g / (m1 + m2)
## Footnote
This formula shows how acceleration is affected by the difference in masses and gravitational force.
54
Calculate the acceleration (a) when m1 = 5 kg and m2 = 25 kg.
a ≈ 6.5 m/s²
## Footnote
This calculation is based on substituting the values of m1 and m2 into the derived acceleration equation.
55
Determine the tension in the cable when m1 = 5 kg and a = 6.5 m/s².
T ≈ 81.7 N
## Footnote
This is calculated using the tension formula T = m1a + m1g with the given values.
56
What is the angular acceleration (α) of the pulley?
α ≈ 65 rad/s²
## Footnote
This is derived from the relationship α = a / r, where a is the linear acceleration and r is the radius.
57
Fill in the blank: The angular velocity (ω) of the pulley 2 seconds after release is _______.
ω ≈ 130 rad/s
## Footnote
This is calculated using the formula ω = ω0 + αt, where ω0 is the initial angular velocity (0 in this case) and t is time.
58
What is the moment of inertia (J) of the wrapped part of the cable?
J = (M * R²) / 2 + m * (l - x) * R²
## Footnote
This equation combines the moment of inertia of the cylinder and the wrapped cable.
59
What is the governing equation for angular acceleration (α) in terms of the unwrapped cable?
α = (m * x * g - T) / J
## Footnote
This equation shows the relationship between the forces exerted on the unwrapped cable and the resulting angular acceleration.
60
Define kinematics in the context of gear systems.
Kinematics is the branch of mechanics describing the motion of bodies without consideration of the masses or forces causing the motion.
61
What is the gear ratio formula for a simple gear train?
Gear ratio = N1 / N2
## Footnote
N1 and N2 are the number of teeth on the input and output gears, respectively.
62
What does an overall gear ratio greater than 1 indicate?
It indicates speed reduction in the system.
63
What is the efficiency (η) of a gear system?
Efficiency is the ratio of output torque to input torque, often expressed as a percentage.
64
Fill in the blank: The conservation law that states no change in momentum occurs when external impulse forces are zero is known as _______.
Conservation of momentum
## Footnote
This principle applies to isolated systems where no external forces act.
65
What is the relationship between impulse and momentum?
Impulse equals the change in momentum.
66
What is the formal definition of Newton's second law in terms of momentum?
The rate of change of momentum is equal to the imposed force.
67
What is the principle of Conservation of Momentum?
The total momentum of a closed system remains constant if no external forces act on it.
68
Define the Coefficient of Restitution (e)
e = Velocity of separation / Velocity of approach
## Footnote
0 < e < 1 indicates an inelastic collision; e = 1 indicates a perfectly elastic collision.
69
What is the formula for Angular Momentum (H)?
H = Jω
70
What does the term 'Impulse' refer to in physics?
Impulse is the change in momentum of an object when a force is applied over time.
71
What is the relationship between Angular Impulse and Angular Momentum?
Angular Impulse = Change in Angular Momentum
72
True or False: Angular momentum changes if external angular impulses are zero.
False
73
What is the equation for conservation of angular momentum before and after an impact?
H_before = H_after
74
What happens to the angular velocity of a system when mass moves closer to the axis of rotation?
The angular velocity increases as the moment of inertia decreases.
75
In the context of a flywheel and clutch system, what does the term 'common shaft speed' refer to?
The speed at which both the flywheel and load rotate together after the clutch engages.
76
Fill in the blank: The moment of inertia (J) of a point mass is calculated using _______.
J = m * r^2
77
What is the formula for calculating work done by a force?
W = F ∙ Δr
78
What is the effect of applying a constant force on the work done?
Work done by a constant force does not depend on the path but only on the starting and finishing points.
79
How is the angular acceleration (α) of a flywheel calculated?
α = ΣM / J
80
What occurs when two discs with different moments of inertia are brought into contact?
Angular momentum is conserved, leading to a new common angular velocity.
81
What is the unit of angular momentum?
kg·m²/s
82
Define 'linear momentum'.
Linear momentum is the product of an object's mass and its velocity.
83
What does the right-hand screw rule define?
The direction of the moment vector.
84
True or False: The moment of inertia increases as mass is distributed closer to the axis of rotation.
False
85
What is the formula for the angular impulse transferred by a clutch?
Angular impulse = Change in angular momentum
86
What is an example of a perfectly elastic collision?
When two billiard balls collide and bounce off without losing kinetic energy.
87
What is the formula for calculating the angular velocity of a system after a person jumps onto a merry-go-round?
H_initial = H_final
88
What factors affect the angular acceleration of a flywheel?
* Applied moment (M)
* Moment of inertia (J)
89
Fill in the blank: The angular momentum of a system remains constant when _______ is conserved.
angular momentum
90
What is 'work' in the context of physics?
The energy transferred to or from an object via the application of force along a displacement.
91
What is the importance of the moment of inertia in rotational dynamics?
It determines how much torque is needed for a desired angular acceleration.
92
What is the expression for work done by a constant force?
𝑊 = ∫12𝐹 ∙ 𝑑𝑟 = 𝐹𝑐𝑜𝑠𝛼(𝑥2 − 𝑥1) + 𝐹𝑠𝑖𝑛𝛼(𝑦2 − 𝑦1)
## Footnote
Work done by a constant force depends only on the starting and finishing points, not the path taken.
93
What is the formula for gravitational potential energy?
V = mgh
## Footnote
Gravitational potential energy is defined as the work done against the gravitational force to elevate a body some distance h above a reference plane.
94
What is the change in gravitational potential energy when moving from height h1 to h2?
Δ𝑉 = 𝑚𝑔(ℎ2 − ℎ1)
## Footnote
This reflects the energy that a body has if held stationary at some height above the reference plane.
95
What is the definition of kinetic energy?
The total work which must be done on the particle in order to bring it from rest to a velocity v
## Footnote
Kinetic energy can also be expressed as T = 1/2 mv^2.
96
What is the change in kinetic energy when accelerating from velocity v1 to velocity v2?
Δ𝑇 = 1/2 m(v2^2 - v1^2)
## Footnote
This formula is derived from the definition of kinetic energy.
97
What characterizes a conservative force?
The work done in moving a particle between two points is independent of the path taken
## Footnote
A conservative system has constant total mechanical energy.
98
What is the work done by the force of gravity?
𝑊 = ∫12𝐹 ∙ 𝑑𝑟 = −𝑚𝑔(𝑦2 − 𝑦1)
## Footnote
Gravitational force is given by 𝐹 = −𝑚𝑔 𝑗.
99
What is the expression for power in terms of work?
P = dW/dt
## Footnote
Power is the rate at which work is done.
100
What is the formula for the velocity of a particle sliding down an inclined plane?
𝑣 = √(2gh)
## Footnote
This is derived from the conservation of energy for a conservative system.
101
What does the work done by the friction force equal?
𝑊 = −𝜇𝑅𝑥
## Footnote
Where 𝜇 is the coefficient of friction, R is the normal force, and x is the distance moved.
102
What is the relationship between gravitational potential energy and kinetic energy?
Loss in gravitational potential energy equals gain in kinetic energy
## Footnote
This principle applies in conservative systems.
103
What is the condition for a system to be classified as non-conservative?
The total mechanical energy within the system is not constant
## Footnote
This typically occurs due to external forces like friction.
104
What is the expression for the total kinetic energy of a coupled system?
T2 = 1/2 mv^2 + 1/2 J(θ̇^2)
## Footnote
J is the moment of inertia and θ̇ is the angular velocity.
105
What is the formula for conservation of momentum in a perfectly elastic collision?
m1u1 = m1v + m2v
## Footnote
This equation conserves momentum before and after the collision.
106
What is the formula for the ratio of masses in a perfectly elastic collision?
m1/m2 = 3
## Footnote
This is derived from the conservation of momentum and kinetic energy equations.
107
What is the significance of the negative sign in gravitational potential energy change?
It indicates that the mass loses potential energy
## Footnote
This occurs when moving downward in a gravitational field.
108
What is the equation relating power to torque in rotational motion?
P = ωM
## Footnote
Where ω is angular velocity and M is the moment.
109
What is the centripetal acceleration for a mass in circular motion?
a = eω²
## Footnote
Where e is the eccentricity and ω is the angular velocity.
110
What is the work-energy principle in a conservative system?
The loss in gravitational potential energy equals the gain in kinetic energy
## Footnote
This principle is foundational in mechanics.
111
What is the role of friction in non-conservative systems?
It converts mechanical energy into thermal energy, reducing total mechanical energy
## Footnote
This results in less mechanical energy available for work.
112
What is the equation representing the balance of forces?
𝑎 + 𝑏 = 0
113
In the ABAmB Plane Single Disk Model, what does the centrifugal force equation represent?
𝐹𝐴 = 𝑚𝑒𝜔²𝑏𝑎 + 𝑏𝐹𝐵 = 𝑚𝑒𝜔²𝑎𝑎 + 𝑏
114
What causes vibrations in the ABAmB Plane Single Disk Model?
Non-zero bearing forces
115
What is the condition for forces equilibrium in Single Disk Balancing with a single mass?
𝑚𝑒𝜔² − 𝑚₁𝑟₁𝜔² − 𝐹𝐴 − 𝐹𝐵 = 0
116
In Single Disk Balancing, what condition must be met for moments equilibrium?
𝑚₁𝑟₁𝜔²(𝑎 − 𝑐) − 𝑚𝑒𝜔²𝑎 + 𝐹𝐵 = 0
117
What does it mean if c=0 in the context of Single Disk Balancing?
Unbalanced condition
118
What are the conditions for achieving static balance in a rotating system?
Zero net out-of-balance force and zero net out-of-balance moment
119
What defines dynamic balance in a rotating system?
Satisfaction of both static balance conditions
120
How is the out-of-balance force calculated when a mass is found out of balance?
Using the formula 𝑚𝑟𝜔²
121
What is the equivalent speed of 65 mph in m/s?
29 m/s
122
What is the formula for angular speed of the wheel?
𝜔 = 𝑣/𝑟
123
What is the out-of-balance force caused by driving the car at 65 mph with a mass of 12.9 kg and eccentricity of 2 mm?
263 N
124
What is the radius of the balancing masses that should be attached to the wheel rims?
203.2 mm
125
What are the characteristics of each disk in the Multiple Discs model?
𝑚𝑖, 𝑟𝑖, 𝜃𝑖, 𝑦𝑖
126
What polygons are used to represent out-of-balance forces and moments?
'mr' polygon for force and 'mry' polygon for moment
127
What must be specified for each balancing mass in Multiple Discs?
* mr value * Angular orientation θ * Position y along shaft
128
What methods can be used to solve balancing problems?
* Graphically * Using trigonometry * Using vertical and horizontal components of vectors
129
What is the mass of the armature in the electric motor example?
100 kg
130
What is the distance between the bearings in the electric motor example?
1.2 m
131
At what speed does the shaft rotate in the electric motor example?
750 rev/min
132
What is the relationship for static balance in the force polygon?
The force polygon is closed
133
What is the net out-of-balance moment calculated in the electric motor example?
296.1 Nm
134
How is the bearing reaction force calculated in the electric motor example?
F = M/d, where M is the net moment and d is the distance between bearings
135
What must be added to each pulley to achieve static and dynamic balance?
Masses at a radius of 100 mm
136
What is the formula to calculate the out of balance force in rotating systems?
There is no specific formula provided, but it involves understanding the forces acting on the bearings during rotation.
137
What is the rotation speed mentioned in the document?
750 rev/min
138
What masses are involved in the system?
* M = 100 kg
* MA = 50 kg
* MB = 40 kg
139
What is the radius at which masses must be added to each pulley?
100 mm
140
What is the resultant magnitude that needs to be added to close the 'mry' polygon?
0.048 kgm²
141
At what angle should the first mass be placed to balance the system?
90° from the vertical
142
At what angle should the second mass be placed to balance the system?
-90° from the vertical
143
What is the value of y given in the document?
1.6 m
144
What is the value of r given in the document?
0.1 m
145
How many kg of mass need to be added to achieve static and dynamic balance?
0.3 kg
146
What concept is emphasized regarding the difference between statically and dynamically balanced systems?
Understanding the difference is crucial for balancing rotating systems.
147
True or False: A system that is statically balanced will always be dynamically balanced.
False
148
What should you attempt to solve after the lecture according to the summary?
All problems on the exercise sheet.
149
What are the causes of out of balance in rotating systems?
Understanding the causes is necessary but specific causes are not detailed in the document.
150
Fill in the blank: The first mass is added in plane C at ______ from the vertical.
90°
151
Fill in the blank: The second mass, identical to the first mass, is added in plane A at ______ from the vertical.
-90°
152
What should you be aware of after the lecture on load characteristics?
How equivalent rotational systems are used to analyse common drive systems and the different load characteristics for various types of load (friction, windage)
## Footnote
Learning outcomes of the lecture
153
How is the gear ratio calculated?
n = N2/N1
## Footnote
Example calculation: n = 100/50 = 2
154
What is the formula to calculate rotational speed?
ω1 = ω1n/ω2
## Footnote
Example calculation: ω1 = 160 rad/s
155
What is the input-output efficiency equation represented by?
η = 0.93
## Footnote
Efficiency is a measure of how much input power is converted to output power
156
Define equivalent rotational system.
A method to express the dynamics of a drive system about an axis using the formula ΣM = L = Jeqα
## Footnote
Where L is the resultant torque, Jeq is the equivalent moment of inertia, and α is the angular acceleration
157
What does the term 'load' refer to in a mechanical system?
Load = Inertia + [Friction(static+dynamics+viscous) + working process + other impedances]
## Footnote
This definition helps in understanding the various components affecting load
158
What are the characteristics of Coulomb friction?
Coulomb friction force is independent of speed and Coulomb friction torque is also independent of rotational speed
## Footnote
This indicates the nature of static and dynamic friction in mechanical systems
159
What is the relationship between viscous friction force and velocity?
Viscous friction force is proportional to relative velocity
## Footnote
Viscous torque is proportional to angular speed
160
What does windage force depend on?
Windage force is proportional to velocity squared
## Footnote
Windage torque is proportional to angular speed squared
161
At rest, what is the value of the load torque?
L = Ls
## Footnote
Represents the static load condition
162
What is the formula for the total moment of inertia of a swing ride?
Jeq = Js + Jr
## Footnote
Where Js is the shaft's moment of inertia and Jr is the total moment of inertia of the riders
163
How to calculate the angular acceleration of the swing ride shaft?
α = (Lmotor - Lload) / Jeq
## Footnote
This equation considers the motor torque and load conditions
164
What are the components of load in a non-steady state operation?
Load curve includes inertial and resistance components
## Footnote
This helps in analyzing the dynamic behavior of the system
165
Fill in the blank: The equivalent moment of inertia is represented by ______.
Jeq
166
True or False: The load-speed characteristics can be scaled using gears with different speed ratios.
True
167
What is the equation for the motor torque-speed curve?
Lmotor = 190000 - 80000ω
## Footnote
This represents the relationship between torque and speed for the motor
168
What is the effect of transmission units like gears in a mechanical system?
They help the system work effectively by matching the torque and load at different rotational speeds
## Footnote
This is crucial in designing mechanical equipment
169
What is the density of the swing's shaft material?
ρ = 3000 kg/m3
170
What is the dynamic Coulomb friction moment around the axis of rotation for the swing ride?
10 kNm
171
What is the proportionality coefficient b2 for the total windage induced torque?
b2 = 11250 N·m/rad²
172
What is the formula for calculating windage torque?
Lwindage = b2ω²
## Footnote
Indicates how windage resistance increases with speed
173
What is the formula for angular acceleration in the context of motor torque?
𝛼 = (180000 − 80000 − 11250 − 10000)/149090
## Footnote
0.53 rad/sec²
174
What do load-speed characteristics depend on in geared systems?
Load-speed characteristics depend on the speed ratios between engine/motor and load
## Footnote
Load requires large torque at low speed, while motor supplies torque at high speed.
175
What is the steady-state load-speed characteristic of a fan attached to an induction motor?
𝐿 = 20 + 0.2𝜔 + 0.025𝜔²
176
How is the load-speed characteristic referred to the motor axis when using a gear with a 3:1 reduction ratio?
𝐿′ = (20 + 0.2𝜔′/3 + 0.025𝜔′²/9)
## Footnote
Efficiency: 𝜂 = 1
177
What is the formula for motor torque-speed characteristics?
𝐿𝑖𝑛𝑝𝑢𝑡 = 5(50𝜋 − 𝜔′)
178
What is the steady-state running speed calculated for the system?
𝜔′ = 150.86 rad/s
179
What is the power supplied by the motor at the running speed of 𝜔′?
𝑃 = 4.7 kW
180
What is the relationship between power required from the engine/motor and the use of gearing?
Using gearing does NOT reduce the power requirements
## Footnote
Inefficiencies in gears increase power requirements.
181
How does windage type load power consumption change with wind speed?
Power consumption ×8 if the wind speed is doubled.
182
What is the equation for power consumed by a windage type load?
𝐿 = 𝑏2𝜔²
## Footnote
Power consumed is: 𝑃 = 𝐿𝜔 = 𝑏2𝜔³
183
What should learners be aware of after the lecture on load-speed and torque-speed characteristics?
• How load-speed and torque-speed characteristics determine steady-state running speed
• How to calculate the power supplied by a motor/engine under steady-state conditions
184
Fill in the blank: The power required from the motor is given by the formula 𝑃 = _______ at running speed 𝜔′.
𝐿𝑖𝑛𝑝𝑢𝑡𝜔′
185
What is the efficiency of the system when referred to the motor axis with a 3:1 reduction ratio?
𝜂 = 1
186
True or False: The load must be referred to the motor/engine axis in load-speed characteristics.
True
