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A Level Further Maths: Simple Harmonic Motion - SHM: Light Damping

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The following example shows how to model light damping by using a second order differential equation. The second-order differential equation modelling light damping is given by: \[ \ddot{x} + 0.5 \dot{x} + 4 x = 0 \] The characteristic equation is: \[ m^2 + 0.5m + 4 = 0 \] Solving for \(m\), we get: \[ m = -0.25 \pm i \sqrt{3.9375} \] Therefore, the general solution to the differential equation is: \[ x(t) = e^{-0.25t} \left( \cos(\sqrt{3.9375} \, t) + 0.1259 \sin(\sqrt{3.9375} \, t) \right) \] where \( A = 1 \) and \( B \approx 0.1259 \) are determined by the initial conditions: Initial displacement \( x(0) = 1 \) m Initial velocity \( \dot{x}(0) = 0 \) m/s The above equation produces a curve that shows light damping in simple harmoinc motion - SHM. If you want to play with it interactively, plea

A Level Further Maths: Demystifying Polar Coordinates through Shapes

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  The above shape is known as cardioid in mathematics. It is produced by the polar equation, r = a(1 + cos(Θ)), where a can be an integer. You can practise a cardioid with the following simulation and see how r and Θ do the magic to produce the shape. Rose Petals You can practise the creation of shapes with the following applet interactively: note the value of n, an odd number,  and the number of petals.   Fermat's Spiral The polar equation of Fermat's Spiral is as follows: r = a√Θ, where a > 0. Fermat's spiral is produced as follows:

The exponential graph

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  The exponential function is a special exponential function; the gradient of any point on the curve is the same as the value of the function at that point.
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Multiple Choice Questions - GCSE Physics - Magnetism & Magnetic Fields - Time: 30 minutes Created and Programmed by Vivax Solutions   Start the Quiz   Submit Quiz

10 Project Ideas for GCSE Computer Science Students

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Great GCSE Project Ideas for Python - computer science Password Generator: Design a password generator that produces secure and memorable passwords based on user-specified criteria. This project will introduce students to concepts like random number generation, string manipulation, and user input validation. They can add their own cipher method with keys to make them very secure. To-Do List Application: Create a simple to-do list application that allows users to add, remove, and mark tasks as completed. This project will teach students about data structures like lists and how to store and retrieve data efficiently. They can use a database, a .csv file or .txt file to store data. Calculator App: Develop a fairly-advanced calculator application capable of performing arithmetic operations like addition, subtraction, multiplication, division and more advanced methods involving trigonometry and statistics. This project will introduce students to variables, operators, and control flow statem

Master the Cosine Rule with Our Interactive and Easy-to-Use Tool

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The cosine rule The cosine rule is a fundamental concept in geometry that can be used to solve a variety of problems. However, it can be difficult to understand and apply, especially for students who are just learning about it. That's why we created a new interactive simulation that makes learning the cosine rule fun and easy. Our simulation allows you to practice using the cosine rule in a variety of situations. You can also see the results of your calculations in real time. We believe that our simulation is the best way to learn the cosine rule. It is interactive, engaging, and easy to use. We hope you enjoy using it! In this blog post, we will provide a brief overview of the cosine rule and how our simulation can help you learn it. We will also discuss some of the features of our simulation that make it so effective. What is the Cosine Rule? The cosine rule is a formula that can be used to find the length of any side of a triangle, given the lengths of the other two sides and th

Demystifying Decimal to Binary Conversion with Python - GCSE and A Level

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Denary to Binary Conversion Denary to binary conversion is a fundamental concept in computer science and digital electronics. It involves transforming a base 10 number, which represents the number system we commonly use, into a base 2 number, which is the language used by computers to store and manipulate information. As far as the computer science students at GCSE and A Level are concerned, it is a vital concept to be mastered. The process of converting from denary to binary involves repeatedly dividing the denary number by 2 and observing the remainders. The remainders represent the digits of the binary number, with a 0 indicating a 0 digit and a 1 indicating a 1 digit. The quotients are then used to perform further divisions, continuing the process until the quotient reaches 0. Here's a step-by-step explanation of the conversion process: Start with the denary number. Divide the denary number by 2. Note the remainder. This will be the least significant bit (LSB) of the binary