Apparatus: Length of light cord, pendulum bob, retort stand and clamp, metre rule, stopwatch (Figure 2.48).
Procedure:
1. Attach one end of the cord to the bob and the other to the retort stand clamp.
Load,W = mg Mass = m kg
Slope = Stiffness, S Slope = S
4π 2
Static deflection = d Square of periodic time = T2
Figure 2.47 Characteristic graphs for mass-spring system
2. Measure and record the length of the cord from the support to the centre of the bob.
3. Displace the bob from its equilibrium position and release it so that it oscillates freely.
4. Note the number of complete oscillations made by the bob in a time of 1 min and calculate the periodic time of the motion.
5. Repeat the procedure with different lengths of the cord until at least six sets of readings have been taken.
6. Calculate the square of the periodic time for each set of read-ings and tabulate the results.
7. Plot a graph of cord length against the square of the periodic time, observe its shape and calculate its gradient.
Theory: The periodic time of a simple pendulum which oscillates with SHM is given by the expression
T¼ 2
ffiffiffil g r
T2¼ 42l g or
l= g 4p2T2
If the pendulum is describing SHM, the graph of pendulum length against the square of the periodic time will be a straight line whose gradient is g/42(Figure 2.49).
Length = I Retort stand
and clamp
Figure 2.48 Arrangement of apparatus
Slope = g 4π2 Pendulum
length = l
Square of periodic time = T2 Figure 2.49 Characteristic graph for simple pendulum
Activity 2.10
The length of a close-coiled helical spring extends by a distance of 25 mm when a mass of 0.5 kg is placed on its lower end. An additional mass of 1.5 kg is then added and displaced so that the system oscillates with an amplitude of 50 mm. Determine (a) the periodic time and frequency of the oscillations, (b) the maximum velocity and acceleration of the mass, (c) the length of a simple pendulum which would have the same periodic time and frequency.
Problems 2.9
1. A mass of 25 kg is suspended from an elastic coiled spring which undergoes a 50 mm increase in length. The mass is then pulled downwards through a further distance of 25 mm and released. Determine (a) the periodic time of the resulting SHM, (b) the maximum velocity of the mass, (c) the maximum acceleration of the mass.
[0.45 s, 0.35 m s 1, 4.9 m s 2] 2. A helical spring is seen to undergo a change in length of 10 mm when a mass of 1 kg is gently suspended from its lower end. A further 3 kg is then added, displaced from the equilibrium through a distance of 30 mm and released. Determine (a) the frequency of the oscillations, (b) the maximum velocity of the mass, (c) the maximum acceleration of the mass.
[2.49 Hz, 0.469 m s 1, 7.34 m s 1] 3. A mass of 6 kg is suspended from a helical spring and produces a static deflection of 6 mm. The load on the spring is then increased to 18 kg and settles at a new equilibrium position. It is then displaced through a further 10 mm and released so that system oscillates freely. Determine (a) the frequency of the oscillations, (b) the maximum velocity and acceleration, (c) the maximum tension in the spring.
[3.7 Hz, 0.233 m s 1, 5.42 m s 2, 247 N]
4. An elastic spring of stiffness 0.4 kN m 1is suspended vertically with a load attached to its lower end. When displaced, the load is seen to oscillate with a periodic time of 1.27 s. Determine (a) the magnitude of the load, (b) the acceleration of the load when it is 25 mm away from the equilibrium position, (c) the tension in the spring when the load is 25 mm away from the equili-brium position.
[6.3 kg, 1.58 m s 2, 71.8 N]
5. A simple pendulum is made from a light cord 900 mm long and a concentrated mass of 2.25 kg. What will be the periodic time of the oscillations when the pendulum is given a small displace-ment and allowed to swing freely? What will be the stiffness of the spring which will have the same periodic time when carrying the same mass?
[1.9 s, 24.5 N m 1] Test your knowledge 2.10
1. How can the stiffness of a spring be determined?
2. How can the circular frequency of a mass-spring system be determined?
3. What is the effect on the periodic time of a mass-spring system of (a) increasing the mass, (b) increasing the spring stiffness?
4. How can the circular frequency of a simple pendulum be calculated?
5. What is the effect on the periodic time of a simple pendulum of (a) increasing the mass of the bob, (b) increasing the pendulum length?
Chapter 3 Mechanical technology
Summary
Mechanical engineering embraces a wide field of activity. Its range includes power generation, land, sea and air transport, manufac-turing plant and machinery and products used in the home and office such as the photocopier, computer printer and washing machine. The term mechatronics is often used to describe systems that incorporate mechanical devices, electrical and electronic cir-cuits and elements of information technology. These are to be found in all of the above areas and it is the aim of this unit to investigate some of the more common mechanical systems and components.Moving parts generally require lubrication and the first part of this chapter examines lubricant types and lubrication systems.
Pressurised systems require seals and gaskets to prevent the escape of lubricants and other working fluids. Rotating parts require bearings and all mechanical systems incorporate fixing devices to hold the various components in position. These will also be examined in this chapter.
A prime purpose of mechanical systems is to transmit power and motion and the various ways in which this can be achieved is investigated. The chapter also provides an overview of hydraulic and pneumatic systems, steam plant, refrigeration and air-conditioning systems and mechanical handling equipment.