Comprehensive Physics Study Guide: Fluids, Torque, and Harmonic Motion

The Buoyant Force Equation: The buoyant force is determined by three primary factors. The equation to represent this is: $F_B = \rho V g$
- $\rho$ (Rho) represents the density of the fluid the object is submerged in. Do not assume the fluid is always water. While the density of water is typically $1000\,kg/m^3$ , certain problems may specify a different density, such as $1250\,kg/m^3$ .
- $V$ represents the volume of the displaced fluid.
- $g$ represents the acceleration due to gravity ( $9.8\,m/s^2$ ).
Factors Affecting Buoyancy: Gravity, the density of the liquid, and the volume of the object/displaced liquid are the only factors that affect the buoyant force.

Fluid Pressure at Depth: The pressure experienced by an object submerged in a fluid depends solely on the depth ( $h$ ) of the object.
- The volume or total size of the container (e.g., a pool) does not affect the pressure.
- Whether a pool is small or massive, if an individual is at the same depth in both, they will experience the exact same pressure.
Flow Rate and Pressure (Bernoulli’s Principle): It is critical to understand the relationship between flow speed and pressure in different cross-sectional areas.
- Smaller Area: The fluid moves faster. In regions where the fluid velocity is high, the pressure is lower.
- Larger Area: The fluid moves slower. In regions where the fluid velocity is low, the pressure is higher.
- Many students incorrectly identify the pressure in these scenarios, so pay close attention to the inverse relationship between speed and pressure.

Torque Definition: Torque ( $\tau$ ) is defined as the product of the force applied and the radius (distance from the pivot point), adjusted for the angle of application: $\tau = F \times r \times \sin(\theta)$
- Angular Dependence: The maximum torque is achieved when the force is applied at a $90^\circ$ angle.
- If the angle is $0^\circ$ , there is no torque.
- Any angle between $0^\circ$ and $90^\circ$ will result in a torque value lower than the maximum.
Equilibrium and the Seesaw Example: In a static equilibrium scenario like a seesaw, torque balance determines placement.
- Force and Distance: There is an inverse relationship between the magnitude of the force and the distance from the pivot (fulcrum). A larger force requires a shorter distance, while a smaller force requires a larger distance to maintain balance.
- Real-world Example: On a seesaw, a small child (smaller force) must sit as far away from the pivot as possible to balance out a parent (larger force) who must sit closer to the pivot.

Spring Graphs: There is a specific graph associated with the unit on springs that must be understood. You must be able to interpret the graph to find the period ( $T$ ).
Period and Frequency Relationship: Once the period is identified from a graph, the frequency can be calculated easily using these inverse relationships: $T = \frac{1}{f}$ $f = \frac{1}{T}$
Period of a Spring: The period of a mass-spring system depends on the mass ( $m$ ) and the spring constant ( $k$ ): $T = 2\pi\sqrt{\frac{m}{k}}$
Period of a Pendulum: The period of a pendulum depends on the length of the string ( $L$ ) and the acceleration due to gravity ( $g$ ): $T = 2\pi\sqrt{\frac{L}{g}}$
- Students must be able to manipulate these equations to solve for any of the variables contained within them.

Forces in Fluids: When an object is in a tank of fluid, multiple forces act upon it. You must be able to draw an FBD showing the following:
- Gravity ( $F_g$ ) acting downward.
- Buoyant Force ( $F_B$ ) acting upward.
Tension Scenarios:
- Hook from Above: If a hook or string is holding the object up from above, the tension force ( $F_T$ ) is directed upward.
- String from Below: If a string is anchored to the bottom of the tank to hold a buoyant object down, the tension force ( $F_T$ ) is directed downward. (Logic: If the string is cut, the object would move upward, meaning the string was pulling down).
Calculations from Equilibrium: Knowing that the system is in equilibrium (net force is zero) allows for the calculation of the density of the fluid ( $\rho$ ) or the volume ( $V$ ) of the object.

Angular Momentum: Review the worksheet regarding people spinning on a platform. Understanding the principles of conservation of angular momentum and how distribution of mass affects spin speed is essential for a specific question involving a figure.
Kinematics Comparison: One question involves comparing an object in free fall versus an object rolling down an incline. Be prepared to discuss the differences or similarities in their motion or energy.

Review of Exam Composition: The instructor noted that there are only 12 questions on this final exam. This is a shorter-than-typical exam length for seniors.
Final Advice: Students were encouraged to pay attention to the specific "pearls" of information dropped, particularly the nuances of fluid pressure and the simple relationships between frequency and period, as these are points where students frequently make mistakes.

Questions & Discussion

Question from student: "Gravity is the only thing that affects the buoyant [force]? Would you agree with that?"
Instructor Response: Yes, in the context of the variables discussed ( $\rho$ , $V$ , and $g$ ), gravity is a key component.
Question from student (Jose): "Jose, you’re not gonna rat me out?"
Instructor Response: This refers to the instructor's decision to only include 12 questions on the final, which is non-standard. The student (presumably Jose) agreed not to tell others about the shortened length of the test.