Further Mechanics

• An object moving in a circular path at constant speed has a constantly changing velocity as velocity has both magnitude and direction, therefore the object must be accelerating, this is known as centripetal acceleration.
• Newton’s first law states that to accelerate, an object must experience a resultant force, therefore an object moving in a circle must experience a force, this is known as the centripetal force, and it always acts towards the centre of the circle.
• Angular speed (ω) is the angle an object moves through per unit time.
• Damping can be used to decrease the effect of resonance, with different types of damping having different effects.
• As the degree of damping increases, the resonant frequency decreases and shifts to the left on a graph.
• The maximum amplitude decreases and the peak of maximum amplitude becomes wider as the degree of damping increases.
• The effects of damping are shown in the graph below, where ζ is the damping ratio, ζ = 1 represents critical damping.
• Angular speed can be found by dividing the object's linear speed (v) by the radius of the circular path it is travelling in (r), or by dividing the angle in a circle (in radians) by the object’s time period (T).
• One radian is defined as the angle in the sector of a circle when the arc length of that sector is equal to the radius of the circle.
• Centripetal acceleration (a) can be found using the formula: r a = r v 2 = ω 2.
• Newton’s second law states that the force (F) on an object is equal to the mass (m) of the object times the acceleration (a) of the object.
• An object is experiencing simple harmonic motion when its acceleration is directly proportional to displacement and is in the opposite direction.
• The equation for simple harmonic motion is: x a = − ω 2.
• An example of a simple harmonic oscillator is the simple pendulum, which oscillates around a central midpoint known as the equilibrium position.
• The pendulum is marked on the diagram by an, which is the measure of displacement, and by an, which is the amplitude of the oscillations, this is the maximum displacement.
• Simple harmonic systems are those which oscillate with simple harmonic motion, examples include: Simple pendulum - A small, dense bob of mass m hangs from a string of length l, which is attached to a fixed point.
• The displacement-time graph of a simple harmonic oscillator follows a cosine or sine curve, with a maximum A, and minimum -A because A and ω are constants.
• The maximum speed and maximum acceleration of a simple harmonic oscillator are A ω and A ω 2 respectively.
• The acceleration-time graph of a simple harmonic oscillator is drawn by drawing the gradient function of the above graph.
• There are two types of mass-spring system, where the spring is either vertical or horizontal, the only difference between these two is the type of energy which is transferred during oscillations.
• For any simple harmonic motion system, kinetic energy is transferred to potential energy and back as the system oscillates, the type of potential energy depends on the system.
• During the oscillations of a simple pendulum, its gravitational potential energy is transferred to kinetic energy and then back to gravitational potential energy and so on.
• When the bob is displaced by a small angle (less than 10°), and let go it will oscillate with SHM.
• For the vertical system, kinetic energy is converted to both elastic and gravitational potential energy, whereas for the horizontal system, kinetic energy is converted only to elastic potential energy.
• For the simple pendulum, the time period is given by the formula: π T = 2 √ l g, where T is time period, l is the length of the string, g is acceleration due to gravity.
• The velocity-time graph of a simple harmonic oscillator is drawn by drawing the gradient function of the above graph, noting that the maximum (ω A) and minimum velocity (-ω A) occurs when x is 0, as expected from the above formula.
• The time period (T) of the oscillations can be measured by measuring the time taken by the pendulum to move from the equilibrium position, to the maximum displacement to the left, then to the maximum displacement to the right and back to the equilibrium position.
• Resonance is where the amplitude of oscillations of a system drastically increase due to gaining an increased amount of energy from the driving force.
• At the amplitude of its oscillations, the system will have the maximum amount of potential energy, as it moves towards the equilibrium position, this potential energy is converted to kinetic energy so that at the centre of its oscillations, the kinetic energy is at a maximum, then as the system moves away from the equilibrium again, the kinetic energy is transferred to potential energy until it is at a maximum again and this process repeats for one full oscillation.
• The total energy of the system remains constant when air resistance is negligible, otherwise energy is lost as heat.
• Resonance can also have negative consequences, such as causing damage to a structure, for example a bridge when the people crossing it are providing a driving frequency close to the natural frequency, it will begin to oscillate violently which could be very dangerous and damage the bridge.
• There are three main types of damping: Light damping, also known as under-damping, where the amplitude gradually decreases by a small amount each oscillation; Critical damping, which reduces the amplitude to zero in the shortest possible time (without oscillating); and Heavy damping, also known as over-damping, where the amplitude reduces slower than with critical damping, but also without any additional oscillations.
• The variation of energy with displacement and time for a simple harmonic system starting at its amplitude is shown in the diagrams above.
• Resonance has many applications such as in instruments, where a long tube in which air resonates, causing a stationary sound wave to be formed; in radio, where they are tuned so that their electric circuit resonates at the same frequency as the desired broadcast frequency; and in swing, where someone pushing you on a swing provides a driving frequency, which can cause resonance if it’s equal to the resonant frequency and cause you to swing higher.
• Damping is where the energy in an oscillating system is lost to the environment, leading to reduced amplitude of oscillations.
• Free vibrations occur when no external force is continuously acting on the system, therefore the system will oscillate at its natural frequency.
• Forced vibrations are where a system experiences an external driving force which causes it to oscillate, the frequency of this driving force, known as driving frequency, is significant.
• If the driving frequency is equal to the natural frequency of a system (also known as the resonant frequency), then resonance occurs.