• Objects at rest stay at rest, and objects in motion stay in motion at a constant velocity unless acted upon by a force.
  • Also known as Galilean frames of reference.
• Non-inertial Frame of Reference: A frame where Newton's laws don't apply. This usually means the frame is accelerating or under strong gravitational influence.

Examples of Inertial Frames:

• Standing still, watching a train go by.
• A spaceship cruising at constant velocity in deep space.
• A car moving at a constant speed on a straight road.

1.2 Rotational Motion 🔄

Let's spin into rotational motion! It's like linear motion, but with angles.

• Rotational Velocity (ω): How fast an object is rotating, measured in radians per second (rad/s). It’s the rotational version of linear velocity.

• Rotational Acceleration (α): How quickly the rotational velocity changes, measured in radians per second squared (rad/s²). It’s the rotational version of linear acceleration.

• Angle (θ): The rotational analog to linear position, measured in radians (rad). Always convert degrees to radians!

• Key Relationships:

  • v = ωr (Tangential velocity = rotational velocity × radius)
  • a = αr (Tangential acceleration = rotational acceleration × radius)
  • ω = Δθ/Δt (Rotational velocity = change in angle / change in time)
  • α = Δω/Δt (Rotational acceleration = change in rotational velocity / change in time)

1.3 Rotational Kinematics

Just like linear kinematics, we have rotational kinematics equations! These are only valid when angular acceleration is constant. 💡

• Period (T): Time for one full rotation.

• ω = 2π/T (Angular velocity in terms of period).

• Centripetal Acceleration (ac): Acceleration towards the center of a circular path.

  • ac = v²/r

Image courtesy of Giphy.

Practice Question

Multiple Choice Questions

1. A wheel is rotating with a constant angular acceleration. Which of the following statements is true? (A) The angular velocity of the wheel is constant. (B) The tangential velocity of a point on the rim of the wheel is constant. (C) The angular displacement of the wheel is changing at a constant rate. (D) The angular velocity of the wheel is changing at a constant rate.

2. A particle moves in a circle of radius r with a constant speed v. What is the magnitude of the particle's acceleration? (A) 0 (B) v/r (C) v²/r (D) v²/2r

Free Response Question

A small ball of mass m is attached to the end of a string of length L. The ball is swung in a horizontal circle at a constant speed v. The string makes an angle θ with the vertical, as shown in the figure below.

[Diagram: A ball of mass m is attached to a string of length L and is swinging in a horizontal circle. The string makes an angle θ with the vertical.]

(a) Draw a free body diagram of the ball, showing all forces acting on the ball. (b) Derive an expression for the tension in the string in terms of m, g, and θ. (c) Derive an expression for the speed of the ball in terms of L, g, and θ.

Answer Key

Multiple Choice

1. (D)
2. (C)

Free Response

(a) Free body diagram should include: * Tension force (T) along the string * Gravitational force (mg) straight down

(b) Vertical components of forces must balance: * Tcosθ = mg * T = mg/cosθ

(c) Horizontal component of tension provides centripetal force: * Tsinθ = mv²/r * r = Lsinθ * (mg/cosθ)sinθ = mv²/(Lsinθ) * v = √(gLsin²θ/cosθ)

2. Forces: Interactions and Laws

2.1 Force Vectors

Forces are vectors! They have both magnitude and direction. 💪

• Vector: A quantity with both magnitude and direction. Examples: force, displacement, velocity, acceleration.

• Scalar: A quantity with only magnitude. Examples: mass, time, speed.

• Net Force: The vector sum of all forces acting on an object.

  • The direction of the net force is the same as the direction of the acceleration (Newton's Second Law: F = ma).

2.2 Force Interactions

Forces are always the result of interactions between objects. No object has force on its own! 🤝

• Forces come in pairs. When you push a wall, the wall pushes back on you.
• Weight force is due to the planet pulling on an object.
• Normal force is due to a surface pushing on an object.

2.3 Newton's Third Law

"For every action, there is an equal and opposite reaction." This is Newton's Third Law in a nutshell. 💥

• Forces always come in pairs, acting on different objects.
• The force magnitudes are equal, and the directions are opposite.
• Important: Gravitational force and normal force are NOT a Newton's Third Law pair, even if they are equal and opposite in some cases. They act on the same object, and the interaction is between different objects.

Practice Question

Multiple Choice Questions

1. A book rests on a table. Which of the following is the reaction force to the weight of the book? (A) The normal force exerted by the table on the book. (B) The force exerted by the book on the table. (C) The gravitational force exerted by the book on the Earth. (D) The force exerted by the Earth on the book.

2. A car collides head-on with a truck. Which of the following is true during the collision? (A) The force exerted by the car on the truck is greater than the force exerted by the truck on the car. (B) The force exerted by the truck on the car is greater than the force exerted by the car on the truck. (C) The force exerted by the car on the truck is equal to the force exerted by the truck on the car. (D) The force exerted by the car on the truck is zero.

Free Response Question

A block of mass m1 is placed on a frictionless horizontal surface and connected to a second block of mass m2 by a light string that passes over a frictionless pulley. The second block hangs vertically, as shown in the figure below.

[Diagram: A block of mass m1 is on a horizontal surface, connected by a string over a pulley to a hanging block of mass m2.]

(a) Draw free body diagrams for both blocks, showing all forces acting on each block. (b) Derive an expression for the acceleration of the system in terms of m1, m2, and g. (c) Derive an expression for the tension in the string in terms of m1, m2, and g.

Answer Key

Multiple Choice

1. (C)
2. (C)

Free Response

(a) Free body diagrams should include: * For m1: Tension (T) to the right, normal force (N) up, and weight (m1g) down. * For m2: Tension (T) up and weight (m2g) down.

(b) Apply Newton's Second Law to each block: * For m1: T = m1a * For m2: m2g - T = m2a * Solve for a: a = m2g / (m1 + m2)

(c) Substitute the expression for a into the equation for m1: * T = m1(m2g / (m1 + m2)) * T = (m1m2g) / (m1 + m2)

3. Final Exam Focus 🎯

Okay, let's focus on what's most important for the exam. Here are the high-priority topics and some last-minute tips:

• Kinematics and Rotational Motion: Make sure you can switch between linear and rotational motion. Know the equations and how they relate to each other.
• Newton's Laws: Understand how forces interact and how to apply Newton's Second and Third Laws. Free-body diagrams are crucial!
• Circular Motion: Be comfortable with centripetal acceleration and how it relates to forces.

Remember, you've got this! Stay calm, take deep breaths, and trust your preparation. You are ready to rock this exam! 🌟