## Machine and Efficiency

### Introduction

Machine is a device that is used to lift larger weights using by applying lesser force. Different machines can be designed to lift larger weights by applying lesser force. A simple example can be a slope. One can raise a larger weight along a slope by applying a force lesser than that of the weight itself. The ratio of the weight lifted and the force required to lift the weight is known as Mechanical Advantage. Its value is always greater than unity.

Another important parameter associated with a machine is its efficiency. Its value is always less than unity. Mechanical Advantage (MA) of a machine has an opposite relationship with the efficiency of the machine i.e. greater the mechanical advantage lesser will be its efficiency. In the case of a slope, the mechanical advantage will be the greatest with the least angle of inclination and vice versa. The reverse will be true with efficiency. Similarly in the case of a pulley system increasing the number of pulleys will lead to an increase in the mechanical advantage at the cost of efficiency. A couple of experiments can be designed to explore the mechanical advantage and efficiency. Two such experiments are designed below.

### Experiment Design

#### Simple slope

This will require a plat surface and a platform with height adjustments to produce an inclined surface with different inclinations. Two known weights to lift along the incline, a spring balance to measure the applied force and a measuring tape to measure the displacement of the weights under the applied force.

What will be measured is the length along the slope and height of the slope to calculate its angle of inclination and displacement of the weights under the applied force.

#### Pulley System

A pulley system with 2, 4, and 6 pulleys and known weights, known weights, a string, a spring balance, and a measuring tape will be required. What will be measured is the force required to lift the weight, change in the length of the string corresponding to a change in the height of the weight.

### Data and Results

The experiment to study the mechanical advantage and efficiency of a slope and a pulley system was carried out in a virtual setting. The experimental data and the relevant calculated parameters pertaining to the slope are presented below in table 1 and table 2 below. The same for the pulley system is presented in table 3 below.

*Table 1: Experimental data and calculated values for the slope.*

*Table 2: Experimental data and calculated values for the slope.*

*Table 3: Experimental data and calculated values for the pulley system.*

### Discussion & Conclusion

The machine performed the most work in trial 6 which corresponds to the greatest mass and the greatest slope of the inclination. It performed the least work in trial 1 that corresponds to the least mass and the least slope. In the case of the pulley system, the greatest work was performed in the first trial and the least work in the second and third trail.

The mechanical advantage was greatest with the least slope of the incline and least with the greatest slope of the incline. In the case of the pulley, the number of pulleys displays a direct relationship with the mechanical advantage. More is the number of pulleys more is the mechanical advantage of the system.

Based on the calculations the six pulley system has the most mechanical advantage. However, the calculations show that the slope is more efficient than the pulleys. This means less energy will be required for the same work output in case of an incline than in the case of pulleys.

Performing this experiment in a hands-on experiment will certainly be not a smooth task as is the case with the virtual experiment. On cannot get a frictionless incline, also to maintain a nearly constant velocity up the slope will not be possible. Measurement of the slope angle, height and distance traveled, etc. will incorporate errors and there will be a larger gap between the hypothesis and the actual result. However, the trend will still be identifiable.

The experimental results confirm well with the hypothesis.

## Thermal Equilibrium

### Introduction

Heat is a form of energy that flows under a temperature gradient. When two bodies at different temperatures are brought in contact with each – other, heat flows from the hotter body (one at the higher temperature) to the colder body (one at a lower temperature) until both the bodies are brought at a common temperature and the two bodies are then said to be in thermal equilibrium. Because the heat flows under a temperature gradient, therefore, rate of heat flow is directly proportional to the temperature gradient. A simple experiment can be designed to carry out the thermal equilibrium experiments. One such simple experiment is described below.

### Design of Experiment

Copper and aluminum blocks of known masses, a connecting rod of copper, thermometers, a heater, and a stopwatch is all that are required. One block can be heated to a certain temperature, keeping the other at room temperature. Then the hot and cold blocks can be brought in contact with each other through the copper rod and the temperature of the two blocks can be continuously measured at a fixed time interval say 30 s and recorded until both the bodies arrive at a common temperature.

### Data and Results

The thermal equilibrium experiments were carried out in a virtual setting. The experimental data and relevant calculated parameters from this experiment are presented in table 1 and table 2 below. The data and results in table 1pertains to the thermal equilibrium while those in table 4 pertaining to the kinetics of the thermal equilibrium or the rate at which the heat is flowing to establish the thermal equilibrium.

*Table 1: Data and results from thermal equilibrium experiments.*

*Table 2: Data and results pertaining to rate of heating/cooling.*

Figure 1 below shows the graph between the rate of heating of the copper block and the temperature difference that drives its heating. This is a linear plot and confirms the direct relationship between the rate of heating and the temperature gradient driving the heating.

### Discussion & Conclusion

From experiments 1 through 3 it can be seen that heating/cooling of cold/hot bodies in thermal contact with each other continues until the thermal equilibrium is established between the two bodies i.e. temperature of the two bodies becomes equal.

Results of experiment 4 confirm that the rate of heating or cooling is directly proportional to the temperature difference driving the heating/cooling. As temperature difference decreases so does the rate of heating/cooling and vice-versa. This is shown in the graph (figure 1 as well)

The findings of the experiments are matching well with the hypothesis derived from theoretical considerations.

## Piston

### Introduction

When gas molecules are slowly allowed into a box enclosed with a frictionless piston and the temperature and pressure remain constant but the volume increases. An increase in volume at constant pressure causes displacement of the piston and the piston while moving against a fixed pressure does some work against the pressure. This work done by the piston is given by the following expression:

W = PΔV = PA*Δd; where Δd is the displacement of the piston at constant pressure due to the inflow of the gas molecules. As per Avogadro’s law, the change in volume is directly proportional to the number of molecules and is independent of the molecular mass of the gas. Therefore, work done by the piston is independent of whether lighter or heavier gas is infused. A simple experiment can be designed to investigate the work done by the piston.

### Design of Experiment

A box-shaped container enclosed by a frictionless piston, with connections for infusing different gases like he and nitrogen. An attachment for the pressure gauge, a thermometer, a flow meter, valves, a suction pump, and a stopwatch.

Different amounts of gases will be added slowly into the container by opening the valve at a fixed flow rate for different time intervals. The pressure of the container and the displacement of the piston will be recorded and the work done will be calculated. This will be done for both lights as well as heavy gas.

### Data and Results

A virtual experiment was carried out to measure work done by a piston against constant pressure when gas molecules are allowed into the container. This experiment was done with a varying number of molecules of light gas and heavy gas molecules.

The experimental data of the displacement of the piston and the work done by the piston is presented below (Table 1) for both light and heavy gas.

*Table 1: Data and calculations for the piston lab experiment.*

The graph between distance and number of the particle is shown in figure 1 below:

The graph between ΔV and W is shown below in figure 2 below:

### Discussion & Conclusion

From the experiments, it is very obvious that displacement of the piston, change in volume of the container, and the work done by the piston are all directly proportional to the number of particles of the gas. This is in line with the hypothesis. Also, the change in volume and work done is independent of the mass of the gas molecules i.e. the relationship is the same irrespective of whether light or heavy species were used.

The findings are in line with theoretical predictions and support the hypothesis.

The relationship between W and ΔV is shown in figure 2 above. This relationship is

W = 23133*ΔV J/m^{3}

Therefore, for 1.5*10^{-23} J work done the required ΔV will be

ΔV = W / 23133 = 1.5*10^{-23} /23133 m^{3} = 6.5*10^{-28} m^{3}

## Simple Harmonic Motion

### Introduction

Simple harmonic motion is a periodic motion in which the force acting on a particle is continuously opposing the displacement of the particle from its mean position. Therefore, this force is termed as restoring force. This force is directly proportional to the displacement of the particle from its mean position and acting in the reverse direction. Restoring force in simple harmonic motion is given by the following expression F = -kx; k is known as spring constant and x is the displacement from the mean position. –ve sign implies that the force is of restoring character. Another important parameter of a simple harmonic motion is its time period. This is the time required for one oscillation.

An example of a simple harmonic motion is a simple pendulum swing at a small amplitude. The time period of this motion is given by T = 2; where l is the length of the pendulum and g is the acceleration due to gravity.

A simple experiment can be designed to calculate the value of g or acceleration due to gravity by using a simple pendulum.

### Design of Experiment

A string, a measuring tape, a stopwatch, and a small steel ball. A simple pendulum will be constructed using the string and the steel ball. The length of the pendulum and the time required to make 10 oscillations will be measured.

### Data and Results

A virtual experiment was carried out to measure the time period of simple pendulums of varying lengths. The experimental data and the calculated values of these experiments are presented in table 1 below:

*Table 1: Data and calculations for the simple pendulum experiment.*

Time period vs length and time period vs square root of length plots are shown in figure 1 below:

### Discussion & Conclusion

The experimental value of the time period of oscillation is matching reasonably well with the theoretically calculated value of the same. The magnitude of relative error is very small. This implies confirmation of the hypothesis.

The graph between time period and length of the pendulum is of parabolic nature. Also, the graph between time period and square root of the length of the pendulum is linear. Both these graphs confirm the relationship established by theoretical treatment.

## Wave

### Introduction

Wave is a form of motion through which energy propagates from one point to another without net movement of the medium. Mechanical waves need a medium to travel while electromagnetic waves do not need any medium and can travel through vacuum. Some important attributes of wave motion are wavelength, frequency, time period, wave velocity etc. Time period of a wave is given by 1/f; where f is frequency of the wave motion. Wave velocity is given by v = *f; where is wavelength of the wave. A simple experiment described below can be used to investigate some of the attributes of a wave motion.

### Design of Experiment

A string, a measuring tape and a stop watch is all that is required. The string will be ties to a stand at one end and disturbance of varying amplitude and wavelength will be created for this experiment. Wavelength, period of oscillation and amplitude will be measured.

### Data and Results

A virtual experiment was carried out to investigate different attributes of a wave motion.

*Table 1: Experimental data and calculated attributes of a string wave.*

### Discussion & Conclusion

In the experiments involving change of frequency; changing frequency led to corresponding changes in the wavelength such that their product or the wave speed remains constant. Therefore, increasing frequency led to decreasing wavelength and vice versa.

Changing amplitude has no effect on the wavelength and also no effect on the wave speed.

From the experimental findings it can be concluded that to produce wave with smallest possible wavelength I will increase the setting of the frequency from 50 to as close as possible near 100. Possible source of error in this experiment is from the measurement of the wavelength and time of for 10 oscillations.