Table of Contents

(The list of parameters can be found below.)

## Motion

### Uniform Linear Motion

Uniform linear motion is a motion that occurs at a constant speed in one direction. It means that an object covers equal displacements in equal intervals of time. In this case, acceleration is zero. This type of motion can be described in the following terms:

v ⇒ velocity [m/s]

s ⇒ displacement [m]

t ⇒ time [s]

### Motion with Uniform Acceleration

This type of motion can be characterized through the changing velocity. The object is moving faster or slower, which means that acceleration does not equal zero and remains constant. Motion with uniform acceleration can be described by **three equations**:

**Displacement-Time**

**Velocity-Time**

a ⇒ acceleration [m/s^{2}]

### Graphs

The types of motion described above can be represented in the following graphs. Uniform linear motion is displayed in red and motion with uniform acceleration is displayed in green.

**Displacement-time graph**

**Velocity-time graph**

**Acceleration-time graph**

### Free Fall

Free fall is a motion of an object where the force of gravity is the only force acting upon it. The force of gravity is a constant parameter with an acceleration a = g = 9.81 m/s, thus free fall refers to a motion with uniform acceleration. With air friction and lift neglected, the following equations remain:

g = 9.81 m/s

h ⇒ height

### Uniform Circular Motion

In this type of motion, an object is moving along a circular path. Since the velocity is a vector, its constantly changing directions balance each other out. Thus, uniform circular motion is defined by the constant sum of velocity. Or to put it simply, if you are driving a car in a circle with the speed of 50 km/h, your acceleration is constant, yet your direction constantly changes.

ω ⇒ angular velocity [1/s]

α ⇒ angular acceleration [1/s^{2}]

n ⇒ rotational speed [1/s]

r ⇒ radius

π ⇒ Pi (approximated as 3.14)

### Periodic Motion

Periodic motion refers to changes of a system or a physical variable based on a fixed position that is repeated approximately or exactly in equal intervals of time. It can be described in the following variables:

T ⇒ period, measured in seconds [s]

f ⇒ frequency, measured in hertz [Hz]

ω ⇒ angular velocity [1/s]

## Momentum and Force

### Newton’s Laws of Motion

**First Law: Law of Inertia**

An object at rest tends to stay at rest, and an object in uniform motion in a straight line tends to stay in that state of motion, unless acted upon by an external force.

This means, under the named circumstances, the velocity of an object remains constant and can be altered only by applying a force.

**Second Law: The Basic Equation of Mechanics**

The change of momentum of an object is proportional to the impulse impressed on the object, and happens along the straight line on which that impulse is impressed.

**Third Law: The Law of Action-Reaction**

Forces always come in pairs; meaning that to every action there is always opposed an equal reaction. When object A exerts a force upon object B (actio), object B exerts a force of the same magnitude upon object A, but the direction of the force is opposite to the direction of the force of object A (reactio).

Action = Reaction

### Momentum

Momentum is a vector quantity, which is used to describe the motion of an object, and its direction is parallel to the motion of that object.

p = m * v

p ⇒ momentum [kg * m/s], [N * s]

m ⇒ mass [kg]

### The Law of Momentum Conservation

The law of momentum conservation states that in an isolated system, which does not have any interaction with its environment, all momentum is constant. So when two objects collide, the total momentum of these two objects before the collision is equal to their total momentum after the collision.

### Forces

Forces deform objects, set them into motion or accelerate their motion. Force is an interaction between objects that changes the energy of an object. The general equation for force is the following:

F = m * a

F ⇒ force [N], [kg * m/s^{2}]

m ⇒ mass [kg]

Different types of forces are displayed in the equations below:

Force |
Equation |
Description |

Force of Gravity (Weight) F_{G} |
F_{g} = m * g |
Force with which the earth attracts another object towards itself (“downward”). Earth acceleration (gravity) is a constant quantity: g = 9.81 m/s^{2} |

Buoyancy (Upthrust) F_{A} |
F F F F |
Upward force in fluids; ρ_{Fl = }density of the fluid [kg/m³]; V = volume of the object placed in the fluid [m³]. |

Spring Force | F_{D} = D * ΔL |
Deflection of a spring D; Spring constant (material-specific constant) [N/m]; ∆L = the length/stretch of a spring [m] |

Normal Force | F_{N} = cos α * F_{G} |
The force exerted perpendicular to the surface of an object. When exerted horizontally, normal force equals weight. α = angle of the inclined surface measured from the horizontal |

Tangential Force | F_{T }= sin α * F_{G} |
Force exerted parallel to the surface of an object. |

Static Friction Force | F_{H} = μ_{H} * F_{N} |
The force exerted by a surface as an object moves across it. It depends on the material and surface quality of the interacting bodies; μ_{H} = static friction coefficient, without unit |

Kinetic (Sliding) Friction Force | F_{Gl} = μ_{G} * F_{N} |
The force that occurs when two objects are moving relative to each other and rub together; μ_{G} = kinetic friction coefficient, without unit |

Rolling Resistance | F_{Roll} = μ_{Roll} * F_{N} |
The force resisting the motion when a body rolls on a surface; μ_{Roll} = rolling resistance coefficient, without unit. Lubricants decrease rolling resistance coefficient, and thus less force is required when moving two colliding bodies. |

Centrifugal Force | F_{Zf} = m * ω * r |
An inertial force that causes an object in a rotating reference frame to move outward away from the axis. r = radius of the circle [m]; M = mass of the object ω = angular velocity [1/s] |

Centripetal Force | F_{ZP} = – F_{Zf} |
A force that is directed from the radius toward the center of the circle; opposed to centrifugal force. |

Coulomb Force | |
It describes how strongly two objects or particles are attracted to each other. It depends on the charge of the objects/particles and the distance between them. ε_{0} = electric field constant; ε_{r} = dielectric constant; Q1, Q2 = charge of the two objects [C]; r = distance to the center of the two objects |

### Impulse

Impulse can be defined as the change over time in momentum caused by an average force. It is defined as follows:

I^{→}^{ }⇒ impulse [kg * m/s]

F_{av} ⇒ average force [N], [kg * m/s2]

p ⇒ momentum

Δt ⇒ duration of impulse

## Torque, Moment of Inertia, Angular Momentum

### Center of Mass

The geometric center often differs from the center of mass, since the latter depends on the density (and therefore the mass) of an object. It can be defined as the centroid of a system with any number of points of the same mass A_{0}, A_{1}, A_{2}… A_{n},:

M_{S }= Center of mass, no unit

The posture of a body determines the type of equilibrium we are in. We distinguish the following types:

- Stable equilibrium: the body comes from the deflected state back to the original posture.
- Unstable equilibrium: after coming back from a deflected state, the body, which had been in equilibrium before, moves further away from the equilibrium state.
- Neutral equilibrium: the body takes a new weight.

### Torque

Torque causes an object to rotate about an axis. It depends on the distance between the axis and the point where the force acts. Force multiplied by force arm equals load multiplied by load arm. Torque can be described in the following equation:

M = r * F

M ⇒ Torque [N * m]

**Example:** Muscles work on joints, which from a physical point of view are their axes. When force is exerted upon them, they create a torque.

### Moment of Inertia

A static/rigid body has resistance. Once a force sets a rigid body in rotating motion, a moment of inertia occurs. It depends on the body’s mass distribution in relation to the axis.

J ⇒ moment of inertia [kg * m^{2}]

r ⇒ axis of rotation

ρ ⇒ mass distribution

### Angular Momentum

Angular momentum refers to what colloquially might be called swirl or spinning. It describes the direction and speed of a rotation about an axis and it increases

- the bigger the mass of the body is;
- the higher the velocity of the body is;
- the longer the distance from the axis is.

It is defined by the following equation:

L = r * p

## Work

W ⇒ work [J], [N * m]

Work is the energy transfer from one object to another so that the second object can be moved or deformed. Like with force, there are several types of work, which are described below.

Work | Equation | Description |

Lifting | W_{H} = F_{G} * h |
Lifting an object at constant speed; h = the height that the object is lifted to |

Acceleration Work |
W F |
Changing the velocity of an object, making it going faster or slower; s = displacement; F_{B}= acceleration |

Pressure-Volume Work | w_{V} = – ∫ p dV |
The volume of a fluid is compressed (V2 < V1) or expanded (V1 < V2) |

## Energy

E ⇒ energy [J], [N * m]

Energy is one of the most fundamental parts of the universe. Energy supports life itself, it allows objects to move despite other forces, pressure to be exerted, substances to be heated, and electric current to flow. The two most essential forms of energy can be described as follows:

Energy | Equation | Description |

Potential Energy | E_{pot} = F_{G} * h |
Energy stored by an object of a certain mass as a result of its position (for example, when it is held at an elevated position); h = height of the object |

Kinetic Energy | E_{kin} = 0,5 * m * v² |
The energy of motion or acceleration; when a resting body is set into motion |

### The Law of Energy Conservation

Since energy can be neither created nor destroyed, energy in a closed system remains constant. A loss of energy occurs only in the sense that it is converted into a different type of energy (motion energy converts into thermal energy, for example) or released (in the form of heat). In this case, the definition of a closed system is essential; a system can be defined as closed if it admits no transfer of energy, information, or mass across it and has no interaction with its environment.

## Power

P ⇒ power [W], [J/s]

Power describes the amount of energy consumed per unit of time, i.e., it is dependent on time. The performed work and the incoming energy are inversely proportional to time. That is, with energy remaining constant, power decreases with time.

### Central Collision

Δp ⇒ collision [(kg * m)/s]

Collision can be defined as a change in momentum over time and can be described in the following equation:

Δp = F * Δt

One can distinguish two types of collision:

- Elastic collision: Kinetic energy is conserved.
- Inelastic collision: A part of kinetic energy is converted into internal energy.

If the centers of gravity of the colliding bodies move along a straight line, such collision is defined as straight-line central collision.

## Pressure

Pressure can be measured with an instrument called manometer. Pressure is the result of a force, exerted upon the surface of an object:

p = F_{N} / A

p ⇒ pressure [Pa], [N/m^{2}]

A ⇒ area of the surface [m^{2}]

V ⇒ volume [m^{3}], [l]

### Boyle-Mariotte Law

The Boyle-Mariotte law describes ideal gases and states that the pressure of a gas is inversely proportional to the volume of a gas. For instance, if you release half of the volume of a gas from an air-compressed tank, the pressure in the tank will be only half as large. It can be described in the following formula:

p ∼ 1 / V

## List of Parameters

t (or t_{0}) |
Time [s] |

v (or v_{0}) |
Velocity [m/s] |

s (or s_{0}) |
Displacement [m] |

a | Acceleration [m/s^{2}] |

g = 9,81 m/s^{2} |
Acceleration of gravity [m/s^{2}] |

h | Height (or falling height) |

α | Angular acceleration |

s | Arc length (= displacement) [m] |

r | Radius [m] |

n | Rotational speed [1/s] |

v | Velocity [m/s] |

T | Period, measured in seconds [s] |

f | Frequency, measured in hertz [Hz] |

ω | Angular velocity (the frequency of circular motion per second) [1/s] |

Φ | An exceeded angle |

p | Momentum [kg * m/s], [N * s] |

m | Mass [kg] |

F | Force [N], [kg * m/s^{2}] |

I → | Impulse [kg * m/s] |

F_{av} |
Average (or, net) force [N], [kg * m/s2] |

M_{s} |
Center of mass, no unit |

M | Torque [N * m] |

J | Moment of inertia [kg * m^{2}] |

r | Axis of rotation |

ρ | Mass distribution |

L | Angular momentum [(kg * m^{2})/s] |

p | Angular momentum of point-mass [(kg * m)/s] |

W | Work (J), [N * m] |

E | Energy [J], [N * m] |

P | Power [W], [J/s] |

Δp | Collision [(kg * m)/s] |

ρ | Pressure [Pa], [N/m^{2}] |

A | Area of a surface [m^{2}] |

V | Volume [m^{3}], [l] |

## Popular Exam Questions on Mechanics

Solutions can be found below the references.

**1. A body made of platinum is immersed in a mercury bath, and a body made of aluminum is immersed in a water bath. The bodies are connected with each other by a thread over deflection pulleys. In the state of equilibrium, the platinum body is fully submerged in mercury, while half of the volume of the aluminum body remains above the water surface.**

**Densities:**

**Platinum: 21.5 g/cm³****Mercury: 13.6 g/cm³****Aluminum: 2.7 g/cm³**

**What is the ratio of volume to volume (aluminum body to platinum body)?**

- 8
- 4.6
- 9.3
- 3.6
- 18

**2. A ****spring is stretched by 50 mm through an attached weight G1. An additional weight G2 with the mass of 30 g stretches the spring an additional 36 mm. With what force does the weight G1 pull the spring?**

- 4 N
- 72 g
- 0.3 N
- 0.7 N
- 42 g

**3. Which of the following statements are false?**

- In uniform acceleration (like free fall), the acceleration increases proportionally to time.
- If acceleration is zero, velocity is also zero.
- If the absolute value of velocity is constant, acceleration is always zero.
- If a body is accelerating perpendicular to its direction of motion with constant acceleration, its path always becomes circular.

- Only b,c,d
- Only c
- Only a
- All, except d
- All of them