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This hub covers everything needed for The 3rd Physics SCT on November 10th (Chapters 7, 8, & 9). Use the navigation bar to jump to specific concepts or seminar problems. Har Har Har Har - Freddy Fazbear
CHAPTER CONCEPTS
CHAPTER 7: ROTATIONAL KINEMATICS
7.1 - RIGID BODIES & ANGULAR QUANTITIES
**Translational Motion** (change of location) vs. **Rotational Motion** (change of orientation). A **Rigid Body** maintains a constant distance between any two of its points.
- **Radian (θ):** Angular measure defined by θ = s / r (arc length/radius). 1 full circle = 2π radians.
- **Angular Velocity (ω):** ωav = Δθ / Δt (Units: rad/s). Related to tangential speed: vt = rω.
- **Angular Acceleration (α):** αav = Δω / Δt (Units: rad/s²). Related to tangential acceleration: at = rα.
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A **Rigid Body** is anything that spins without bending or changing shape, like a bowling ball or a spinning tire.
**Angular Velocity (ω)** is just **how fast something is spinning** (like **RPM**s on a car).
**Angular Acceleration (α)** is **how quickly the spinning speed is changing** (speeding up or slowing down).
CHAPTER 8: ROTATIONAL DYNAMICS
8.1 - TORQUE τ
A **Torque (τ)** is the rotational equivalent of force, measuring the tendency to cause angular acceleration.
The magnitude is given by τ = rF sin(θ) or τ = Fd, where d is the moment arm. The SI unit is **Newton-meter (N·m)**.
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**Torque** is the **"twist" or turning force**. It's what makes a wrench turn a bolt. The further away you push, the more torque you get.
8.2 - MOMENT OF INERTIA (I)
The **Moment of Inertia (I)** is the rotational resistance to changes in angular velocity, analogous to mass (m) in linear motion.
For a system of masses: I ≡ Σmr².
**Rotational 2nd Law:** Στ = Iα.
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**Moment of Inertia** is an object's **resistance to spinning**. The heavier the object and the more its mass is spread out (like a bicycle wheel rim), the harder it is to start or stop it from spinning.
CHAPTER 9: SOLIDS & DEFORMATION
9.1 - STATES OF MATTER & ELASTICITY
Matter is classified as **Solid, Liquid, or Gas** (predominant on Earth) or **Plasma** (predominant in the universe).
The relationship between the deforming force (Stress) and the resulting deformation (Strain) is summarized by Hooke's Law: **Stress = Elastic Modulus × Strain**.
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- **Solid:** Keeps its shape.
- **Liquid:** Takes the shape of the container.
- **Gas:** Fills the entire container.
**Elasticity** means an object can be stretched or squished, but it will **go back to its original shape** when the force is removed (like a spring).
CHAPTER 9: FLUIDS (PRESSURE, BUOYANCY, FLOW)
9.2 - DENSITY, PRESSURE, & DEPTH
The **Density (ρ)** of a substance is mass per unit volume: ρ ≡ M/V (Units: kg/m³).
Pressure variation with depth (h): **P = P0 + ρgh**.
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**Density** is **how much "stuff" is packed into a space**. Something dense, like a rock, is heavy for its size.
**Pressure** is **Force** pushing on an **Area** (like the difference between being poked with a finger vs. a pin).
9.3 - BUOYANT FORCE & ARCHIMEDES'S PRINCIPLE
**Archimedes's Principle** states that the buoyant force (B) equals the **weight of the fluid displaced**.
The buoyant force is calculated as: **B = ρfluid g Vdisp**.
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The **Buoyant Force** is the **upward push** that a liquid gives to an object, which is why things can float.
**Archimedes's Principle** explains that this **upward push is exactly equal to the weight of the water** that the object pushes out of the way.
9.4 - FLUIDS IN MOTION (CONTINUITY & BERNOULLI)
- **Equation of Continuity:** The product of area and speed is constant: **A1v1 = A2v2**.
- **Bernoulli's Equation:** Conservation of energy in a moving fluid: **P + ½ρv² + ρgh = Constant**.
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**Continuity** means that if water flows into a narrow pipe (smaller area), it **must speed up**. (A narrow river flows faster than a wide one.)
**Bernoulli's Principle** means that for a moving fluid, where the speed is high, the **pressure is low** (and vice versa). This is why airplane wings generate lift.
SEMINAR PROBLEMS (INPUT ANSWERS & CHECK HINTS)
Q.17 (CH. 7): SWINGING GIRL (Tension)
Problem:
A **25.0 kg** girl on a swing is moving at **5.00 m/s** at the lowest point. She holds a horizontal rope attached to a pole. She moves in a circle of radius **0.800 m**. Determine the **force (T)** exerted by the horizontal rope on her arms (in Newtons).
&implies; ANSWER (T in N, round to nearest whole number):
Q.18 (CH. 7): ARCHEOLOGIST VINE (Breaks?)
Problem:
An **85.0 kg** archeologist swings from a **10.0 m** vine at **8.00 m/s** at the bottom. The vine breaks at **1,000 N**. **Does he make it across the river without falling in?** (Yes/No)
&implies; ANSWER (Yes/No):
Q.19 (CH. 7): VERTICAL CIRCLE (Tension & Accel.)
Problem:
A **0.500-kg** object swings in a vertical circle (**r=2.00 m**). At **θ=20.0°**, **v=8.00 m/s**. Find (a) the **tension** in the string (in N).
&implies; ANSWER (Tension T in N, round to one decimal):
Q.20 (CH. 8): PULLEY TENSIONS (Massive Pulley)
Problem:
Two masses, **M = 5.0 kg** and **m = 2.0 kg**, are connected by a light string over a solid disk pulley (M' = 1.0 kg). Find the magnitude of the **tension TM** in the rope supporting the heavier mass (M) (in N). Assume the rope does not slip.
&implies; ANSWER (TM in N, round to one decimal):
Q.21 (CH. 9): WOOD CUBE BUOYANCY
Problem:
A cube of wood ($s=20.0 \text{ cm}$, ρwood = 650 kg/m³) floats on water (ρwater = 1000 kg/m³). Find the mass of lead ($m_{lead}$) that should be placed on the cube so that the top of the cube will be just level with the water surface (in kg).
&implies; ANSWER (mlead in kg, round to one decimal):
Q.22 (CH. 9): BERNOULLI FLOW
Problem:
A liquid (ρ = 1650 kg/m³) flows through a horizontal pipe. In Section 1: A1=10.0 cm², v1=2.75 m/s, P1=1.20 × 10&sup5; Pa. In Section 2: A2=2.50 cm². Calculate the smaller section's **pressure (P2)** in Pascals (Pa).