Real-world examples of first-order ODEs derived from physical principles

📚 Quick Study Guide

• 🍎 Newton's Law of Cooling: Describes how an object's temperature changes over time in relation to its surroundings. The equation is: $\frac{dT}{dt} = -k(T - T_s)$, where $T$ is the object's temperature, $t$ is time, $T_s$ is the surrounding temperature, and $k$ is a constant.
• 🌊 Simple Harmonic Motion: Models the oscillatory motion of a mass attached to a spring. The equation is: $m\frac{d^2x}{dt^2} + kx = 0$, which can be rewritten as a first-order system.
• 🔌 RL Circuit: Describes the current in a circuit containing a resistor and an inductor. The equation is: $L\frac{dI}{dt} + RI = V(t)$, where $I$ is the current, $t$ is time, $L$ is the inductance, $R$ is the resistance, and $V(t)$ is the voltage source.
• 🧪 Radioactive Decay: Models the decay of radioactive substances. The equation is: $\frac{dN}{dt} = -λN$, where $N$ is the number of radioactive atoms, $t$ is time, and $λ$ is the decay constant.
• 💧 Torricelli's Law: Describes the rate at which fluid drains from a container. The equation is: $\frac{dh}{dt} = -k\sqrt{h}$, where $h$ is the height of the fluid, $t$ is time, and $k$ is a constant dependent on the orifice size and gravity.

Practice Quiz

1. A hot cup of coffee is placed in a room with a constant temperature of 20°C. Which ODE best describes the cooling of the coffee, according to Newton's Law of Cooling?

   1. $\frac{dT}{dt} = k(T - 20)$
   2. $\frac{dT}{dt} = -k(T + 20)$
   3. $\frac{dT}{dt} = -k(T - 20)$
   4. $\frac{dT}{dt} = k(20 - T)$

2. In a simple harmonic motion system, what does the term 'kx' represent in the ODE $m\frac{d^2x}{dt^2} + kx = 0$?

   1. Damping force
   2. External force
   3. Spring force
   4. Gravitational force

3. An RL circuit has an inductance of 2H and a resistance of 4Ω. If the voltage source is 12V, what is the differential equation describing the current I(t)?

   1. $2\frac{dI}{dt} + 4I = 12$
   2. $4\frac{dI}{dt} + 2I = 12$
   3. $6\frac{dI}{dt} = 12$
   4. $2\frac{dI}{dt} - 4I = 12$

4. A radioactive substance decays at a rate proportional to its amount. If the decay constant is 0.05, which ODE models this decay?

   1. $\frac{dN}{dt} = 0.05N$
   2. $\frac{dN}{dt} = -0.05N$
   3. $\frac{dN}{dt} = N - 0.05$
   4. $\frac{dN}{dt} = -N - 0.05$

5. According to Torricelli's Law, what happens to the rate of fluid draining from a container as the height of the fluid decreases?

   1. The rate increases.
   2. The rate decreases.
   3. The rate remains constant.
   4. The rate oscillates.

6. Which of the following real-world phenomena can be modeled using a first-order ODE?

   1. Projectile motion
   2. Population growth with limited resources
   3. Planetary orbits
   4. Fluid dynamics in complex systems

7. Which parameter affects the rate of cooling in Newton's Law of Cooling?

   1. The object's mass
   2. The object's volume
   3. The temperature difference between the object and its surroundings
   4. The object's material