Equations of Motion, Dislocating the Object - Lab Report Example

Equation of Motion Name Institutional Affiliation Date Table of Contents Introduction 3 Theory 3 Linear Motion 3 Circular Motion 4 Deriving Equations under Linear Motion 4 Deriving the Equations under Circular Motion 8 Experiment 1: Linear Motion 13 Equipment and Procedure 13   18 Experiment 2: Circular Motion 20 Experiment and Procedure 20 Conclusion 24 References 25 Introduction Equations of motions describe how a body moves in relation to time. The main variables of a physical system in linear motion are distance and time. Linear motion describes an object that is moving on a straight line, for example, a trolley moving in a straight line on a table. In rotational or circular motion, an object is assumed to be rotating on a specific axis. Bodies that may be described to be in a rotational motion are like the merry- go- round and the earth. Thus, this paper seeks to design experiments, collect the relevant data, analyze the results and come up with equations of motion that describe both linear and angular motions. Theory Linear Motion For linear motion, when an object is moving in a straight line, its speed in a given direction is velocity (Asada, Futamase & Hogan 2011). Velocity increased with time is acceleration. In circular motion, speed in a circle is angular velocity and when angular velocity is increased, centripetal acceleration increases. An object in motion moves at a specific speed. Speed is a scalar quantity, it doesn’t have direction. Speed that has direction is known as velocity denoted by (v, u) .Such quantities that have direction are known as vector quantities. Circular Motion The force that we experience when a body is going around a curve is what is known as centripetal acceleration, for example, a vehicle taking a corner (Gibilisco 2007). It is the centripetal acceleration that keeps the body in motion when it is going round in a curve. This force is what keeps the stars, the earth, the moon and even the sun in their orbits and makes them to move in curves and not in straight lines. In circular motion, there exists a relationship between centripetal acceleration and angular velocity, in that angular velocity is the square root of its function. As centripetal acceleration increases, the velocity of the object increases in a relation that is equal to the square root of the radius by gravity (Walecka & Walecka 2010). Deriving Equations under Linear Motion Since speed is the distance moved for a specific period of time then; Further, distance that has definite direction is known as displacement. Therefore, displacement in time (t) is velocity. Hence, If we consider an object that starts at velocity (u) and accelerates constantly at acceleration (a) to final velocity (v) after some time (t), then its acceleration (a) is Where; the acceleration in meters per second square (m/s2) final velocity of the object in meters per second (m/s) initial velocity in meters per second (m/s) time the object takes in seconds (s) Simplifying the equation This is the first equation of motion. The diagram below shows the relation between velocity (m/s) and time (t) Figure 1: Velocity - Time Graph Total displacement= average velocity× time From the diagram in Figure 1, the area under the graph is what gives displacement as shown in Figure 2. Figure 2: Area under Velocity Time graph Thus, the area under the graph = Displacement (s) = 2 Hence, 2 Second equation of motion Therefore Substituting in Equation 2 gives: ) t Thus, = 2- u2 So Third equation of motion Deriving the Equations under Circular Motion In circular motion, the object is moving on a circular path as shown and the above variables are denoted by different letters but the equations are generally the same as those for linear motion (Morrison 2010). Figure 3: Movement of an Object in Circular Motion The angular distance turned through is denoted by The initial angle ϴ0 Angular velocity ῳ Initial angular velocity ῳ0 Angular acceleration is denoted by and can be derived by comparing two similar triangles with two equal radii as shown.  Change in v/v= Where Vector –vo=Vector vf =v The small change in time, the arc length is s to c. We know distance is rate × time, we replace the c with The subscript represents the centripetal acceleration for objects on circular motion. If you have an object moving at a constant speed where is the circumference and is the time for one revolution, also known as periodic time   .   If we substitute the expression for in the equation   Substituting this expression for c into the equation for centripetal acceleration,  Yields Experiment 1: Linear Motion Equipment and Procedure Equipment Ticker tape, ticker timer, a runway, trolley, power supply, a ruler, lab clamp and jumper wire  Procedure Using a one meter long ticker tape, pull it through a ticker timer. The two should be aligned in a straight line and same level. Attach the tape to the trolley and pull it parallel to the runway Figure 3. Figure 4: Experimental Setup  Set the ticker timer to 10 cycles per second. Place the trolley close to the ticker timer and give it a gentle push once the timer has been switched on. It should be noted at this point that in 10 cycles per second, for one cycle is 0.1 seconds and thus, the distance between two dots on the tape is 0.1 sec.  Measure the distance which is the length of the distance between two ticker marks. Make about measurements consecutively while avoiding the initial several dots. Using the first equation, calculate the speed for each consecutive dot. Record the results of Dx, Dt, and vav in the table below and calculate the average velocity by averaging all the vav Using the same set up to connect the trolley to a mass through a pulley as shown below  Release the trolley once you have switched on the ticker timer and hold it before it knocks the pulley. Record the distance between the dots as Dx1, Dx2 till the last one. Avoid the first few dots. Calculate velocity of the distance between the two consecutive dots. Vi=Dxi/ Dti Where is To determine other values and fill the table below. The table below has been completed for you. Table 1: Table Showing Value for Distance and Time Draw graphs of Distance verses Total time using the values obtained in the Table above to obtain Figure 4. Figure 5: Graph of Distance in Centimeters verses Time in seconds       This graph gives us the instant accelerated motion at any given time. Figure 6: Instant velocity per unit Time Graph Average velocity in a time interval for the total time taken is the total average velocity of the trolley in the whole journey. To get instantaneous velocity at a certain moment in the graph, mark the midpoint at the top of the bars and connect the point to the best line of fit. Acceleration Verses time  Finally, plot graph of acceleration against time using the data obtained. Below is an example using the data that was obtained from the above experiment. Determine acceleration by calculating the gradient of the velocity verses time graph. Compare this average value of accelration obtained with the theoretical values from the lecturers notes and state the reason for the difference in the value. Therefore, from the experiments above, an object is moving in a straight line, its speed in a given direction is velocity and the relationship between the variables is; 2 Velocity increased with time is acceleration and the two are related as shown: v2=u2+2as In circular motion, speed in a circle is angular velocity. When angular velocity is increased, centripetal acceleration increases hence in all these cases the three equations of motion. Experiment 2: Circular Motion Experiment and Procedure Equipments Weights, string and plastic tubing, stop watch, rubber stoppers Procedure First before starting, swing the rubbers on the string with the brass weight giving the centripetal force. The plastic tubing is then held by your hand to enable the string to slide freely on the tube without creating too much friction. Hold the tube in a vertical position while swinging the stopper in a circle at a speed while keeping the radius constant. It is important to note that increase in speed will make the radius taken by the ball to increase while slower speeds will decrease the radius. Maintain the speed constant and practice such that the speed is at a point of increasing but remaining constant. Hold the stopper at that speed as your partner times you using a stop clock to measure time for ten revolutions. When the stopper is at high speeds, the string will tend to be almost parallel in relation to the floor but at lower speeds, it is pulled down several degrees. Note that in this case you will consider the distance as the radius of the distance from the holder to the middle point of the rubber stopper. Attach a small tape to the string near the end of the plastic holder to keep the radius at some value while the stopper is moving round at a speed that is constant. This radius can also be kept constant by observing its position in relation to the position held Make sure the tape does not make the string to stop and sliding on the tube smoothly. Start to swing the rubber at a constant radius and measure the time it takes for exactly ten revolutions. On average, measure three measurements of the time taken for that radius and average for good results . You should prepare a table like the one below. d is the total distance in ten revolutions tav is the average time taken for the ten revolutions. The Table below is the completed table with the results From the results above, about six to eight values that of a specific known radius take the mass of the stopper and the weight of the brass in Newton. Using relevant formulas find the distance travelled in the ten revolutions, average time for the ten revolutions and the speed. It is a linear graph that goes through the origin, the slope of the graph is what gives us centripetal acceleration. It will be found that the acceleration is the same for the data at the points. The reason behind this is that larger radii allow greater speeds for the same acceleration. This is the reason as to why a car can move faster when a turn is gradual. The acceleration from the graph represents the average value for centripetal accelerations that are theoretically the same. Ideally, centripetal acceleration is the same for all the points. This is because longer radii allow greater speed for the same acceleration. The acceleration from the graph is average for the stopper to cause centripetal acceleration. You can compare this force and express it as a percentage difference. From the analysis above it is clear that there exists a relationship between centripetal acceleration and angular velocity, whereby angular velocity is the square root of its function. As centripetal acceleration increases, the velocity of the object increases in a relation that is equal to the square root of the radius by gravity. That is; Conclusion In the laboratory, there are many variables we cannot control. In the experiment, the controlled variables are the mass of the stopper that is undergoing centripetal acceleration in a spinning motion. Others include time taken for one revolution but this may vary for so many reasons. The mass hanging is an independent variable which depends on the weights added. To eliminate tension force components, we assume the stopper rotates at a perpendicular distance. References Top of Form Bottom of Form Top of Form Bottom of Form Top of Form Bottom of Form Top of Form Asada, H., Futamase, T., & Hogan, P. A. 2011. Equations of motion in general relativity. New York: Oxford University Press. Gibilisco, S. 2007. Advanced physics demystified. New York, McGraw-Hill. Morrison, J. C. 2010. Modern physics for scientists and engineers. Burlington, MA, Academic Press/Elsevier. Walecka, J. D., & Walecka, J. D. 2010. Advanced modern physics: theoretical foundations. Singapore, World Scientific. Bottom of Form Read More

Using the first equation, calculate the speed for each consecutive dot. Record the results of Dx, Dt, and vav in the table below and calculate the average velocity by averaging all the vav Using the same set up to connect the trolley to a mass through a pulley as shown below  Release the trolley once you have switched on the ticker timer and hold it before it knocks the pulley. Record the distance between the dots as Dx1, Dx2 till the last one. Avoid the first few dots. Calculate velocity of the distance between the two consecutive dots.

Vi=Dxi/ Dti Where is To determine other values and fill the table below. The table below has been completed for you. Table 1: Table Showing Value for Distance and Time Draw graphs of Distance verses Total time using the values obtained in the Table above to obtain Figure 4. Figure 5: Graph of Distance in Centimeters verses Time in seconds       This graph gives us the instant accelerated motion at any given time. Figure 6: Instant velocity per unit Time Graph Average velocity in a time interval for the total time taken is the total average velocity of the trolley in the whole journey.

To get instantaneous velocity at a certain moment in the graph, mark the midpoint at the top of the bars and connect the point to the best line of fit. Acceleration Verses time  Finally, plot graph of acceleration against time using the data obtained. Below is an example using the data that was obtained from the above experiment. Determine acceleration by calculating the gradient of the velocity verses time graph. Compare this average value of accelration obtained with the theoretical values from the lecturers notes and state the reason for the difference in the value.

Therefore, from the experiments above, an object is moving in a straight line, its speed in a given direction is velocity and the relationship between the variables is; 2 Velocity increased with time is acceleration and the two are related as shown: v2=u2+2as In circular motion, speed in a circle is angular velocity. When angular velocity is increased, centripetal acceleration increases hence in all these cases the three equations of motion. Experiment 2: Circular Motion Experiment and Procedure Equipments Weights, string and plastic tubing, stop watch, rubber stoppers Procedure First before starting, swing the rubbers on the string with the brass weight giving the centripetal force.

The plastic tubing is then held by your hand to enable the string to slide freely on the tube without creating too much friction. Hold the tube in a vertical position while swinging the stopper in a circle at a speed while keeping the radius constant. It is important to note that increase in speed will make the radius taken by the ball to increase while slower speeds will decrease the radius. Maintain the speed constant and practice such that the speed is at a point of increasing but remaining constant.

Hold the stopper at that speed as your partner times you using a stop clock to measure time for ten revolutions. When the stopper is at high speeds, the string will tend to be almost parallel in relation to the floor but at lower speeds, it is pulled down several degrees. Note that in this case you will consider the distance as the radius of the distance from the holder to the middle point of the rubber stopper. Attach a small tape to the string near the end of the plastic holder to keep the radius at some value while the stopper is moving round at a speed that is constant.

This radius can also be kept constant by observing its position in relation to the position held Make sure the tape does not make the string to stop and sliding on the tube smoothly. Start to swing the rubber at a constant radius and measure the time it takes for exactly ten revolutions. On average, measure three measurements of the time taken for that radius and average for good results . You should prepare a table like the one below. d is the total distance in ten revolutions tav is the average time taken for the ten revolutions.

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