Free Body Diagrams and the Atwood Machine: Measurement of Gravity

Conceptual Overview: Lesson five of the introductory physics course (AP Physics 1 Review) focuses on Free Body Diagrams (FBDs) as a mathematical modeling tool. The central theme is the "Gravitational Field Trip," utilizing FBDs to precisely measure the acceleration due to gravity on Earth ( $g$ ).
Standard Value of Gravity: The acceleration due to gravity near Earth's surface is approximately $9.81\,m/s^2$ . In many simplified physics problems, this value is rounded to $10\,m/s^2$ .

Gravity as a Fundamental Force: Gravity is one of the four known fundamental forces. Despite its impact on a planetary scale, it is considered a very weak force proportional to its mass.
The Four Fundamental Forces:
- Gravity: Responsible for the attraction between masses.
- Electromagnetic Force: Responsible for magnetic properties (attraction/repulsion) and static electricity (e.g., a balloon sticking to hair).
- Strong Nuclear Force: Involved in holding atomic nuclei together.
- Weak Nuclear Force: Involved in radioactive decay.
Strength Comparison: A magnet smaller than a pea can pick up a metal nail, overcoming the entire gravitational pull of the Earth. This demonstrates that gravity is significantly weaker than the electromagnetic force.

Definition: An FBD is a simplified diagram representing all the forces acting on a single, isolated object (referred to in physics as a "body").
The Point Particle Model: In FBDs, complex objects (like a cow or a bottle of nail polish) are modeled as a single point particle to simplify the calculation of vectors.
Application Examples:
- Falling Object (e.g., Nail Polish): If air resistance is neglected, the only force is gravity ( $F_g = m \times g$ ), represented by a downward arrow.
- Projectile Motion (e.g., Soccer Ball): For a ball in flight (neglecting air resistance), the only force acting on it is gravity pointing downward. There is no horizontal force acting on the ball after it has left the kicker's foot. The FBD for a projectile is identical to that of a free-falling object.

Air Resistance Defined: A force created by air pushing against a moving object. It acts in the opposite direction of motion and increases as speed increases.
Dynamics of a Skydiver:
1. Initial Fall: Speed is low, air resistance is negligible, and the net force is downward ( $a = g$ ).
2. Acceleration Phase: As speed increases, the upward force of air resistance ( $F_{air}$ ) increases.
3. Terminal Velocity: Eventually, $F_{air}$ becomes equal in magnitude to the force of gravity ( $F_g$ ). At this point, the sum of the forces is zero: $\sum F = F_{air} - F_g = 0$
4. Result: The skydiver ceases to accelerate and falls at a constant speed called terminal velocity.
The Magnus Effect: A phenomenon where air curves the trajectory of a spinning spherical object (e.g., a curving soccer ball) due to pressure differences created by the spin.

Scale Mechanics: A scale does not measure mass directly; it measures the contact force (the normal force) compressing its internal springs.
Scenario 1: Constant Velocity or Static: If an elevator is stationary or moving at a constant speed, acceleration ( $a$ ) is zero. $\sum F = F_{scale} - F_g = m \times 0 = 0$ The scale reading equals the actual weight ( $mg$ ).
Scenario 2: Upward Acceleration: If the elevator accelerates upward at $a = 2\,m/s^2$ with a mass of $58\,kg$ :
- $F_g = 58\,kg \times 10\,m/s^2 = 580\,N$
- $\sum F = m \times a = F_{scale} - F_g$
- $58\,kg \times 2\,m/s^2 = F_{scale} - 580\,N$
- $116\,N = F_{scale} - 580\,N$
- $F_{scale} = 696\,N$
- Conclusion: The person feels heavier because the scale/floor must push up with more force to provide the upward acceleration.
Scenario 3: Downward Acceleration/Deceleration: When an elevator slows down at the top, it decelerates (net force is downward). The springs in the scale compress less, and the perceived weight decreases.
The Vomit Comet: A zero-gravity plane achieves weightlessness by controlling its descent so that the downward acceleration equals $g$ , resulting in a net perceived weight of zero.

History: Invented in 1784 by George Atwood to study constant acceleration.
Mechanism: A pulley with a rope and two different masses ( $m_1$ and $m_2$ ) on either side. It allows for slow, measurable acceleration.
Key Principles:
1. Both masses are connected by a non-stretching rope, so they share the same magnitude of acceleration ( $a$ ).
2. The tension ( $T$ ) in the rope is uniform throughout.
Mathematical Derivation for $g$ :
- For the descending mass ( $m_1$ ): $-m_1 \times a = T - m_1 \times g$
- For the ascending mass ( $m_2$ ): $m_2 \times a = T - m_2 \times g$
- Equating the tension ( $T$ ) from both equations leads to: $g = a \times \frac{m_1 + m_2}{m_1 - m_2}$
Experimental Data:
- $m_1 = 79\,kg$ , $m_2 = 65\,kg$ (human-sized experiment).
- Distance ( $d$ ) = $2\,m$ , Time ( $t$ ) = $2.15\,s$ .
- Using kinematics ( $d = \frac{1}{2}at^2$ ): $a = 0.87\,m/s^2$ .
- Calculated $g = 8.9\,m/s^2$ .
Error Analysis: The discrepancy (vs. $9.81\,m/s^2$ ) is attributed to pulley friction, weight of the cable ( $\approx 2\,kg$ ) changing during the fall, and measurement sensitivity of the masses.

Variable Gravity: Gravity is not perfectly uniform across Earth. It varies between $9.76\,m/s^2$ and $9.83\,m/s^2$ .
Factors:
- Altitude: Gravity weakens as distance from the Earth's center increases.
- Density: Areas over dense deposits (e.g., uranium mines) exhibit higher local gravitational pull.
- NASA Mapping: NASA has created maps of "gravity anomalies" showing these variations across the globe.

Dialogue regarding the construction of the human-sized Atwood machine:

Diana: "I'm here with Thea. Behind us is the human sized Atwood machine. Thea made this. Do you remember, like, what the specifications were?"
Thea: "So the specifications for this were basically it had to be human size, support Diana safely, and be as frictionless as possible because we essentially, it's a pulley system. We're offsetting two weights, and we want to make sure to minimize variables."

To solve for acceleration ( $a$ ) quickly given two masses: $a = g \times \frac{m_1 - m_2}{m_1 + m_2}$
Example Analysis: If $m_1 = 2 \times m_2$ :
- $a = g \times \frac{2m_2 - m_2}{2m_2 + m_2}$
- $a = g \times \frac{m_2}{3m_2}$
- $a = \frac{1}{3}g$
Conceptual Insight: Gravity only works on the difference in the masses, but it must accelerate the sum of the masses.

Net Force Problems: Always start by drawing a Free Body Diagram (FBD).
Component Problems: When dealing with multiple objects/components, look for common variables (like tension or shared acceleration).
Principle of Equivalence: In standard free fall, things fall at the same rate because gravitational mass is equal to inertial mass. The Atwood machine allows us to bypass this for observation.