Author Capstone Axis

Rotational Motion About a Fixed Axis – Student Notes Rotational Motion About a Fixed Axis – Quick-Read Notes When an object spins around a fixed line (the axis), every point on it shares the same angle \( \theta \). Because only that one angle matters, the motion has just one degree of freedom. :contentReference[oaicite:1]{index=1} 1  Big […]

Moment of Inertia: The Rotational “Mass” When an object spins around a fixed line (axis), every bit of matter in it moves in a circle. A piece at distance \(r_i\) from the axis has speed \(v_i = r_i\omega\), where \(\omega\) is the angular speed. Its kinetic energy is \[ k_i = \tfrac12 m_i v_i^2 =

Equilibrium of a Rigid Body – Student-Friendly Notes 1  Mechanical equilibrium in plain words A rigid body is in perfect balance when it neither speeds up in a straight line nor starts spinning faster or slower. In symbols: \(\displaystyle \sum \vec F_i = \vec 0\)  — no net push ⇒ no linear acceleration :contentReference[oaicite:0]{index=0} \(\displaystyle \sum \vec

Torque & Angular Momentum – Friendly Notes 1. Why Torque Matters Push a door near the hinge and nothing happens; push the outer edge and it swings open. That everyday difference introduces torque – the twist that forces create when they act away from the pivot. :contentReference[oaicite:0]{index=0} 2. Defining Torque (Moment of Force) Vector form: \( \boldsymbol{\tau}

Angular Speed & Linear Speed: Quick, Friendly Notes 1. What is Angular Speed (ω)? Imagine a point on a spinning wheel. As the wheel turns through a tiny angle dθ in a tiny time dt, its angular speed is \( \displaystyle \omega = \frac{d\theta}{dt} \). :contentReference[oaicite:0]{index=0} Angular speed is a vector. Point your right-hand thumb

Vector Product (Cross Product) of Two Vectors Why we care The “cross” of two vectors helps us build moment of force (torque) and angular momentum—two pillars of rotational motion.:contentReference[oaicite:0]{index=0} Definition For two vectors \(\mathbf a\) and \(\mathbf b\), their vector product is \(\mathbf c = \mathbf a \times \mathbf b\). Magnitude: \( |\mathbf c| = ab\sin\theta

Linear Momentum of a System of Particles Think of momentum as the “oomph” a moving object carries. For one particle we write \( \mathbf{p}=m\mathbf{v} \), where m is the mass and \( \mathbf{v} \) is the speed-and-direction (velocity) vector. :contentReference[oaicite:0]{index=0} 1. Quick Refresher: One Particle Momentum formula: \( \mathbf{p}=m\mathbf{v} \). Newton’s second law in its “momentum outfit”:

Motion of Centre of Mass (COM) Imagine n tiny chunks of matter, each with mass mi and position ri. Their common balance point – the centre of mass – sits at R and obeys $$M\mathbf R \;=\; m_1\mathbf r_1 + m_2\mathbf r_2 + \dots + m_n\mathbf r_n \tag{6.7}$$ :contentReference[oaicite:0]{index=0} Speed of the COM: Differentiate once to get $$M\mathbf V =

Centre of Mass: Quick Notes Imagine tracking a whole bunch of particles by following just one special point that represents the “average” location of their mass. That point is the centre of mass (CM). Here’s how to find it and why it helps. 1. Two-Particle System Place the two masses on an x-axis at positions x1 and x2. The

Systems of Particles & Rotational Motion – Friendly Notes 1. Why talk about “systems of particles” at all? Every real object has size, so treating it as a single point sometimes breaks down. The way around that is to picture the object as a system of many particles. When we do that, two ideas dominate: