Higher Advanced Mathematics of Mechanics SQA

This subject is broken down into 26 topics in 3 modules:

  1. Forces, Energy and Momentum 9 topics
  2. Straight Line, Periodic and Parabolic Motion 5 topics
  3. Mathematical Techniques for Mechanics 12 topics
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This page was last modified on 28 September 2024.

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Advanced Mathematics of Mechanics

Forces, Energy and Momentum

Applying impulse, change in momentum and conservation of momentum

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Applying impulse, change in momentum and conservation of momentum

Impulse and Change in Momentum

  • Impulse is defined as force times time: it is the product of the average force exerted on an object and the time interval over which the force acts.
  • In mathematical terms, impulse is also defined as the change in momentum of an object when a force is applied.
  • Momentum is a vector quantity as it has direction as well as magnitude.
  • In physics, momentum is usually denoted by the letter 'p', and it is calculated by multiplying the mass of an object (m) by its velocity (v). Therefore, p=mv.
  • Momentum is conserved in a closed system where external forces are not at play. This is known as the conservation of momentum.

Applying the Concept of Impulse and Change in Momentum

  • The concept of impulse can be applied to understand how the velocity of an object changes when a force is applied.
  • For example, when a football is kicked, the shoe applies a force to the football over a short period of time. This force changes the football’s velocity, and hence its momentum.
  • The greater the impulse, the greater the change in momentum. If the force is applied for a longer period of time, the change in the object’s momentum will be more significant.
  • In an impact between two objects, the total momentum before the impact is equal to the total momentum after the impact (assuming no external forces are at work). This concept is central to the principle of conservation of momentum.

Conservation of Momentum

  • The principle of conservation of momentum states that in a closed system, the total linear momentum of the system remains constant if no external forces act on it.
  • In a collision, the total momentum of the colliding objects is the same before and after the collision.
  • Therefore, if you have two objects, their combined momentum before the collision equals their combined momentum after the collision.
  • This principle has significant practical applications in real-world physics including, for example, the study of vehicle collisions or the performance of rocket propulsion systems.

Course material for Advanced Mathematics of Mechanics, module Forces, Energy and Momentum, topic Applying impulse, change in momentum and conservation of momentum

Advanced Mathematics of Mechanics

Straight Line, Periodic and Parabolic Motion

Applying the concept of simple harmonic motion (SHM)

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Applying the concept of simple harmonic motion (SHM)

Understanding Simple Harmonic Motion (SHM)

  • Simple Harmonic Motion is a special type of periodic motion where an object moves back-and-forth along a straight line path.
  • It is characterised by its amplitude, period, frequency, and phase.
  • The object in SHM moves under a restoring force which is directly proportional to its displacement but in the opposite direction.
  • The mathematical expression that describes SHM is x = A cos(wt + φ), where x is the displacement, A is the amplitude (maximum displacement), w is the angular frequency, t is the time, and φ is the phase (initial) angle.

Amplitude, Period and Frequency

  • The amplitude (A) is the maximum distance that the object moves from the equilibrium position, in either direction.
  • The period (T) is the time taken for the object to complete one full cycle of the motion.
  • The frequency (f) is the number of cycles that the object completes in unit time. It is the reciprocal of the period, i.e., f = 1/T.
  • The frequency of SHM is also related to the angular frequency w, where w = 2πf.

Restoring Force and Energy in SHM

  • The restoring force (F) in SHM is the force that pulls the object back toward the equilibrium position. It is given by F = -kx, where k is the force constant and x is the displacement.
  • SHM involves a continuous transfer of energy from kinetic (due to motion) to potential (due to object's position).

Applications of SHM

  • Examples of SHM include the motion of a simple pendulum, spring-mass system, and oscillations in electric circuits.
  • Understanding the properties and characteristics of SHM is important in a variety of fields, including physics, engineering, and astronomy.
  • Problems involving SHM can be solved by applying the principles of mechanics and using the mathematical expressions associated with SHM.

Remember, simple harmonic motion is a fundamental concept that helps us understand the natural phenomena around us. A deep understanding of its properties will help you solve complex problems involving periodic and oscillatory motions.

Course material for Advanced Mathematics of Mechanics, module Straight Line, Periodic and Parabolic Motion, topic Applying the concept of simple harmonic motion (SHM)

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