Simple Pendulum, Power, and Impulse: Study Notes for College Physics

Simple Pendulum

Introduction to the Simple Pendulum

The simple pendulum is a classic example of periodic motion, consisting of a mass (called the bob) attached to a string of length L that swings under the influence of gravity. The motion is governed by the restoring force due to gravity, which causes the pendulum to oscillate back and forth about its equilibrium position.

• Key Terms: Length (L), Angle (\theta), Height (h), Period (T), Frequency (f).

• Equation of Motion: For small angles, the restoring force leads to simple harmonic motion.

Energy in a Simple Pendulum

The total mechanical energy of a simple pendulum (ignoring air resistance and friction) is conserved and is the sum of its potential and kinetic energies:

• Conservation of Energy:

$mgh + \frac{1}{2}mv^2 = \text{constant}$

• Maximum Speed: At the lowest point, all potential energy is converted to kinetic energy:

$mgh = \frac{1}{2}mv_{\text{max}}^2$

$v_{\text{max}} = \sqrt{2gh}$

Equation of Motion and Small Angle Approximation

The motion of the pendulum can be described by the following differential equation:

$\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0$

• For small angles (\theta in radians), $\sin\theta \approx \theta$ (small angle approximation).

• This simplifies the equation to simple harmonic motion:

$\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$

Period of a Simple Pendulum

The period is the time taken for one complete oscillation. For small angles, the period is independent of mass and amplitude:

$T = 2\pi\sqrt{\frac{L}{g}}$

• Example: For L = 1.00 m, T ≈ 2.01 s.

• Applications: Pendulum clocks use this property for accurate timekeeping.

Power

Definition and Importance of Power

Power is the rate at which work is done or energy is transferred. It is a crucial concept for understanding the efficiency and performance of machines and biological systems.

• Definition: $P = \frac{dW}{dt}$

• Units: 1 watt (W) = 1 joule/second (J/s)

• Instantaneous Power: $P = \vec{F} \cdot \vec{v}$

• Average Power: $\overline{P} = \frac{W}{\Delta t}$

Examples and Applications of Power

• Example 1: A 60-kg jogger runs up a 4.5 m high flight of stairs in 4.0 s. Calculate the power output.

$\overline{P} = \frac{mgh}{\Delta t} = \frac{60 \times 9.8 \times 4.5}{4.0} = 660\ \text{W}$

• Example 2: How many horses are needed to pull a 700 kg buggy at 3 m/s against a 400 N friction force? (1 hp = 746 W)

$P = Fv = 400 \times 3 = 1200\ \text{W} = 1.61\ \text{hp}$

• Example 3: Power required to lift an elevator (1800 kg total) at 3.00 m/s against a 4000 N friction force.

$P = (Mg + f)v = (1800 \times 9.8 + 4000) \times 3.00 = 6.49 \times 10^4\ \text{W}$

Impulse and Momentum

Momentum and Impulse

Momentum is the product of an object's mass and velocity. Impulse is the product of the net force and the time interval over which it acts, and it equals the change in momentum.

• Momentum: $\vec{p} = m\vec{v}$

• Impulse: $\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$

• Units: N·s or kg·m/s

Impulse-Momentum Theorem

The impulse-momentum theorem states that the impulse delivered to an object is equal to its change in momentum:

$\vec{F} \Delta t = \Delta \vec{p} = m\vec{v}_2 - m\vec{v}_1$

• Graphical Interpretation: The impulse is the area under the force vs. time curve.

Impact Force and Stopping Time

For a given change in momentum, increasing the time of impact reduces the average force experienced. This principle is crucial in safety engineering (e.g., airbags, crumple zones).

• Example: A 2000 kg car changes velocity from -15.0 m/s to +2.60 m/s. Calculate the average force for stopping times of 0.05 s and 0.5 s.

$\Delta p = 2000 \times (2.6 - (-15.0)) = 35,200\ \text{kg·m/s}$

For $\Delta t = 0.05$ s: $F_{\text{ave}} = \frac{35,200}{0.05} = 704,000$ N

For $\Delta t = 0.5$ s: $F_{\text{ave}} = \frac{35,200}{0.5} = 70,400$ N

Applications: Airbags and Safety

Airbags increase the stopping time during a collision, thereby reducing the force on passengers. This significantly improves safety in car accidents.

• Key Point: Force by dashboard (shorter stopping time) is much greater than force by airbag (longer stopping time).

Summary Table: Key Equations and Concepts

Concept	Equation	Units
Period of Pendulum	$T = 2\pi\sqrt{\frac{L}{g}}$	s
Power	$P = \frac{dW}{dt}$, $P = \vec{F} \cdot \vec{v}$	W (J/s)
Impulse	$\vec{J} = \vec{F}_{\text{avg}} \Delta t$	N·s or kg·m/s
Momentum	$\vec{p} = m\vec{v}$	kg·m/s
Impulse-Momentum Theorem	$\vec{J} = \Delta \vec{p}$	N·s or kg·m/s