Dynamics Cheatsheet

Cheatsheet Content

### Dynamics Overview Dynamics is the branch of mechanics concerned with the study of forces and their effect on motion. It combines kinematics (the description of motion) with the causes of that motion (forces). #### Key Concepts: - **Linear Momentum:** The product of an object's mass and its velocity. - **Impulse:** The change in momentum of an object, often resulting from a force applied over a period of time. - **Newton's Laws of Motion:** Fundamental principles governing the relationship between forces and motion. ### Linear Momentum - **Definition:** Linear momentum ($\vec{p}$) is a vector quantity defined as the product of an object's mass ($m$) and its velocity ($\vec{v}$). $$\vec{p} = m\vec{v}$$ - **Units:** kilogram-meter per second (kg m/s) or Newton-second (N s). - **Conservation of Momentum:** In an isolated system (where no external forces act), the total linear momentum remains constant. - For a collision between two objects: $$m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$$ where $\vec{u}$ represents initial velocities and $\vec{v}$ represents final velocities. - **Elastic vs. Inelastic Collisions:** - **Elastic Collision:** A collision where both momentum and kinetic energy are conserved. The relative speed of approach equals the relative speed of separation. $$(\vec{u}_1 - \vec{u}_2) = -(\vec{v}_1 - \vec{v}_2)$$ - **Inelastic Collision:** A collision where momentum is conserved, but kinetic energy is not (some kinetic energy is converted to other forms like heat or sound). If objects stick together, it's a perfectly inelastic collision. - **Relationship with Force:** Newton's second law can be expressed in terms of momentum: $$F = \frac{\Delta p}{\Delta t}$$ This means force is the rate of change of momentum. ### Newton's Laws of Motion These three laws form the foundation of classical mechanics. 1. **First Law (Law of Inertia):** An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced external force. - **Inertia:** The property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force. Mass is a measure of inertia. 2. **Second Law (Law of Acceleration):** The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of the acceleration is in the direction of the net force. $$\vec{F}_{net} = m\vec{a}$$ - **Key Points:** - Force and acceleration are always in the same direction. - Solves problems involving $F=ma$ for constant mass systems. 3. **Third Law (Law of Action-Reaction):** For every action, there is an equal and opposite reaction. - When object A exerts a force on object B, object B simultaneously exerts an equal in magnitude and opposite in direction force on object A. - These forces act on different objects and therefore do not cancel each other out. ### Impulse - **Definition:** Impulse ($\vec{J}$) is the change in momentum of an object. It is also equal to the product of the average force ($\vec{F}$) acting on an object and the time interval ($\Delta t$) over which the force acts. $$\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$$ - **Units:** Newton-second (N s) or kilogram-meter per second (kg m/s). - **Force-Time Graphs:** - The area under a force-time graph represents the impulse. - This is particularly useful for forces that are not constant. $$\vec{J} = \int \vec{F}(t) dt$$ - **Applications:** - Understanding impact forces (e.g., crumple zones in cars increase $\Delta t$ to reduce $\vec{F}$). - Analyzing sports actions (e.g., a golfer following through increases $\Delta t$ to maximize $\Delta p$). ### Explaining the use of Force-Time Graphs & ICT - **Force-time graphs** are visual representations of how a force changes over a period. The area under the graph provides the impulse, which is crucial for analyzing impacts and variable forces. - **ICT Simulations:** Computer simulations are invaluable tools for: - **Demonstrating elastic and inelastic collisions:** Visualizing how velocities and kinetic energies change in different collision types. - **Analyzing complex systems:** Exploring scenarios that are difficult or dangerous to set up physically. - **Data collection and analysis:** Quickly generating and processing data for force-time graphs or momentum changes. ### Applications & Tools - **Trollies:** Often used in experiments to demonstrate momentum conservation and collisions due to their low friction. - **Force Metre:** A device used to measure the magnitude of a force. - **Stop Watch:** Used to measure time intervals, essential for calculating impulse and acceleration. - **Slotted Masses:** Used to vary the mass of objects in experiments, allowing for investigation of the relationship between force, mass, and acceleration. - **Light Gates/Motion Sensors:** Electronic devices used to accurately measure time, velocity, and acceleration in experiments. - **Metre Rule/Tape Measure:** Used for measuring distances and displacements.