Understanding Tension with Examples

Tension

• Tension represents the force exerted by a string, rope, cable, or similar object on another object.
• It's a pulling force, acting to draw the object.
• Tension always aligns with the direction of the string or cable.
• In an ideal string, the tension remains uniform across all points.

Example 1: Block Hanging from a String

• A block of mass $m$ hangs from a string
• Tension of the string is calculated by first constructing a free body diagram
• This diagram identifies the forces: the weight $\vec{P} = m\vec{g}$ and the Tension $\vec{T}$
• Then apply Newton's equation to the system $\sum \vec{F} = 0 = m\vec{a}$
• Solving the equation $\vec{T} + \vec{P} = 0$ leads to $\vec{T} = -\vec{P}$.
• Magnitude of Tension: $T = P = mg$, indicating the tension equals the weight.

Example 2: Block Pulled by a String with Force F

• A block of mass $m$ dragged by string with a force $F$
• Free body diagram includes: weight ($\vec{P} = m\vec{g}$), normal force ($\vec{N}$), tension ($\vec{T}$), and applied force ($\vec{F}$).
• Netwon's Equation $\sum \vec{F} = m\vec{a}$, resulting in the equation $\vec{T} + \vec{P} + \vec{N} + \vec{F} = m\vec{a}$
• X-axis component: $T \cos \theta + F = m a_x$
• Y-axis component: $T \sin \theta + N - P = 0$
• Assuming no friction $F=0$, $T \cos \theta = m a_x$
• Tension is expressed as $T = \frac{m a_x}{\cos \theta}$

Example 3: Two Blocks Connected by String over a Pulley

• Two blocks with masses $m_1$ and $m_2$ connected by a string over a pulley
• Free body diagram incorporates:
  • Block 1: Weight ($\vec{P_1} = m_1 \vec{g}$) and tension ($\vec{T}$)
  • Block 2: Weight ($\vec{P_2} = m_2 \vec{g}$) and tension ($\vec{T}$)
• Newton's Equations:
  • Block 1: $\sum \vec{F_1} = m_1 \vec{a_1}$, which becomes $T - m_1 g = m_1 a_1$
  • Block 2: $\sum \vec{F_2} = m_2 \vec{a_2}$, resulting in $T - m_2 g = -m_2 a_2$
• If pulley is ideal: $a_1 = a_2 = a$
• Tension calculation $T = \frac{2 m_1 m_2 g}{m_1 + m_2}$

Bernoulli's Principle

• An increase in fluid speed occurs simultaneously with a decrease in its pressure or potential energy
• Lift is the force that opposes the weight of an aircraft

Airfoil

• Is the shape of a plane wing that is designed so that air flows faster over the top than the bottom

Pressure

• Difference in speed creates a difference in pressure
• The higher pressure below the wing pushing it upwards

Lift

• The faster the airplane moves, the more lift is generated

Applications of Bernoulli's Principle

• Atomizer: Used in perfume bottles and spray paint to create a fine mist.
• Chimney: Helps draw smoke upward by reducing the pressure at the top.
• Carburetor: Mixes air and fuel in an internal combustion engine
• Pitot tube: Measures fluid speed, like air or water.
• Sail of a sailboat: Uses wind to generate lift for boat propulsion.

Bernoulli's Equation

• States that the total energy of a fluid flowing in a closed system is contant
• Written as: $P + \frac{1}{2} \rho v^2 + \rho g h = constant$
  • $P$ is the pressure of the fluid
  • $\rho$ is the density of the fluid
  • $v$ is the velocity of the fluid
  • $g$ is the acceleration due to gravity
  • $h$ is height of the fluid above a reference point

Example

• Water flows horizontally and speed increases from 2.0 m/s to 4.0 m/s
• If the pressure at the low speed section is 100 kPa
• $100,000 Pa + \frac{1}{2} (1000 kg/m^3) (2.0 m/s)^2 = P_2 + \frac{1}{2} (1000 kg/m^3) (4.0 m/s)^2$
• Where $P_2 = 94,000 Pa$, the section in the high speed section is 94 kPa

Anderson-Darling Test

• A goodness-of-fit test used to assess if a data sample comes from a specific distribution
• Enhancement of the Kolmogorov-Smirnov test, better at detecting differences in the distribution tails

Hypothesis

• Null hypothesis is that the data follows the specified distribution.
• The alternate is that it does not come from the distribution

Test statistic

• The equation $A^2 = -n - \sum_{i=1}^{n} \frac{(2i - 1)}{n} [\ln(F(Y_i)) + \ln(1 - F(Y_{n+1-i}))]$, is used to determine the test statsitc
  • $n$ is the sample size
  • $Y_i$ are the ordered data
  • $F$ is the cumulative distribution function of tested distribution

Critical Value

• The threshold depends on the significance level ($\alpha$) and the distribution being tested.
• These can be found in tables or statistical software.

Decision

• Reject the null hypothesis if test statistic exceeds the critcal value

Example In Python

• The Anderson-Darling test is performed on a random data sample and is used to determine if the values come from a normal distrubution
• If the printed test-statistic exceeds the critical value the null hypothesis is rejected

Assumptions

• Data is independent and identically distributed
• Assumes the tested distribution is continuous

Advantages

• Better detection of differences at the tail of the distribution
• Can be applied to various distribution types

Disadvantages

• More mathematically intricate than the Kolmogorov-Smirnov test.
• requires specifying the distribution from which the sample is evaluated against

Related Tests

• Kolmogorov-Smirnov test
• Chi-squared test

Poisson Process, General Definition

• A stochastic model used for the number of events occurring over a time interval

Definition

• The definition of a stochastic process ${N(t), t \geq 0}$ as a Poisson process with rate $\lambda > 0$ requires that
  • $N(0) = 0$, starts at zero events.
  • Independent increments: The number of events in disjoint time intervals are independent.
  • Stationary increments: The distrubution in the number of events depends only on the length of the interval.
  • $P(N(t+h) - N(t) = 1) = \lambda h + o(h)$
  • $P(N(t+h) - N(t) \geq 2) = o(h)$
    • Where $o(h)$ such that $\lim_{h \to 0} \frac{o(h)}{h} = 0$

Assumptions

• Implies events occur one at a time
• Probability of even occurring is proportional to the length of the interval.

Properties

• $N(t) \sim Poisson(\lambda t)$, i.e. $P(N(t) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}$
• The inter-arrival times are exponentially distributed with rate $\lambda$.
• The $n$th arrival time has a Gamma distribution with shape $n$ and rate $\lambda$.

Thinning and Superposition

• Analyzes processes with a rate $\lambda$ to independently classify Type I and Type II events

Thinning

• Suppose each event of a Poisson process with rate $\lambda$ is independently classified as type I with probability $p$ and type II with probability $1-p$
• $N_1(t)$ and $N_2(t)$ be the number of type I and type II =$N_1(t)$ and $N_2(t)$ are independent Poisson at rates $\lambda p$ and $\lambda(1-p)$ respectively

Superposition

• $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates $\lambda_1$ and $\lambda_2$
• $N(t) = N_1(t) + N_2(t)$ with rate $\lambda_1 + \lambda_2$.