UNESCO-NIGERIA TECHNICAL & VOCATIONAL EDUCATION REVITALISATION PROJECT-PHASE IINATIONAL DIPLOMA IN MECHANICAL ENGINEERING TECHNOLOGY MECHANICAL ENGINEERING SCIENCE[STATICS] YEAR I- SE MESTER I PRACTICALS Version 1: December 2008 1 PROGRAMME: MECHANICAL ENGINEERING SCIENCE (STATICS-MEC 111) COURSE SPECIFICATION PRACTICAL CONTENT TABLE OF CONTENTS Week 1 1. Experiment No 1: Parallelogram of Forces Week 2 2. Week 3 3. Week 4 4. Experiment No.4: Calculation of coefficient of Friction Week 5 5. Experiment No.5: Friction on an Inclined Plane Week 6 6. Experiment No.6: Angle of Friction Week 7 7. Experiment No.7: Sliding Friction Week 8 8. Experiment No 8: Principle of Moments Week 9 9. Experiment No 9: The Pivot (or beam) balance Week10 10. Experiment No.10 Forces in Frame Structures Week11 11. Experiment No. 11 Finding the point where an Object’s mass acts Experiment No 3: Polygon of Forces Experiment No 2: Triangle of Forces 2 Week12 12. Experiment No.12 mo r e s c i e n c e o f b a l a n c e Week13 13. Experiment No. 13 to verify the laws of limiting friction Week14 14. Experiment no.14 to verify Lami’s theorem Week15 15 Experiment no.15 to verify triangle law ii 3 they are equivalent to single force (the resultant) acting somewhere in between them. It can be shown that when three such forces are balanced (that is in equilibrium). cord. weights. their lines of action all meet at a point. metal ring. Apparatus: Force board. Until the sledge moves. and drawing-pins 4 . The sledge will move in a direction between the ropes along the line of their resultant force. pulleys. drawing-paper. their lines of action meet at a point. the resultant of two forces in the same plane at an angle can be found by a graphical method called the Parallelogram of Forces. Using this fact. it will pull back against the ropes with a single horizontal force equal and opposite to the resultant of the two rope forces.EXPERIMENT NO 1: PARALLELOGRAM OF FORCES Teaching Element: When two forces act on a body in different directions in one plane. and hence to show that the resultant of two forces can be found using the Parallelogram of Forces. Student Objectives: The object of this experiment is to test that when three non-parallel forces in the same plane are in equilibrium. An example of this is when a sledge is pulled by two horizontal ropes spread at an angle. 2. What is the direction of the resultant? 3. 8. and join OC. Are the resultant and the central weight equal in magnitude and opposite in direction? 4. the point at which the mark is being made. (Care must be taken to ensure that the eye is placed level with. Note the values of the three weights. and mark a length along OA to represent the force which acted on the corresponding cord. Remove the drawing-sheet.) 5. Using the chosen scale. This is the resultant of the forces in the cords OA and OB. 5 . State the theorem on which the method is based. Choose a suitable scale. and indicate beside the line the weight at the end of the cord. and directly in front of. 9. Repeat for line OB. Make a mark at the centre of the ring and one under each cord. Complete the parallelogram AOBC. 6. 4. Pin a sheet of drawing-paper to the board. What is the magnitude of the resultant found by this method? 2. find the force represented by OC. 7. Conclusion 1. 3.Parallelograms of Forces Method 1. Fix the pulleys in any position and suspend weights so that the cords are at rest. Put an arrowhead on each line to show the sense of the force. Join the central mark (O) to each of the other three by straight lines. 3. Student Objectives: The object of this experiment is to test that three non-parallel forces in equilibrium can be represented by a triangle of forces.1 Triangle of Forces Method 1. Because of this the forces can be represented by a force diagram called the Triangle of Forces.EXPERIMENT NO 2: TRIANGLE OF FORCES Teaching Element: When three forces in the same plane act in different directions on a stationary body their lines of action meet at a point. 6 . Note the values of the three weights. Pin a sheet of drawing-paper to the board. 2. Fix the pulleys in any position and suspend weights so that the cords are at rest. This can be used to find the size of two of the forces when the third force is known. provided that the direction or line of action of the forces is known. Apparatus: as in Experiment No. from which two of the forces can be found when the third force is known. 2. 6. and on it mark a length AC to represent the force Q to scale. 8. Join BC. Put an arrowhead on each line to show the sense of the force. Join the central mark (A) to each of the other three by straight lines. the point at which the mark is being made. Is BC parallel to the cord carrying load R? Does BC represent the force R drawn to scale? Can the forces be represented by the sides of a triangle? What do you notice about the arrow-heads on the forces in the triangle ABC? State the theorem known as the “Triangle of Forces. Make a mark at the centre of the ring and one under each cord. 7. and directly in front of. Produce. Conclusions 1. and indicate beside the line the weight at the end of the cord.) 5. 3.” 7 . 4. 9. 5. Along the cord AD mark a length AB to represent the force P to scale. beyond A.4. Remove the drawing-sheet. the line representing Q. (Care must be taken to ensure that the eye is placed level with. and whereas Tri means three. two unknowns can be found in magnitude or direction if the remaining information is known. 2. so that each is in line with the centre of the ring. 3. it is necessary to find the forces acting in each member so that the frame can be made strong enough to withstand the maximum loads exerted upon it. and drawing-pins. cord. The Polygon of Forces is frequently employed to find such forces and deals with each joint in turn. Adjust the cords slightly if necessary.EXPERIMENT NO 3: POLYGON OF FORCES Teaching Element: In the design of pin-jointed plane structures such as girders. drawing-paper. bridges and roof trusses. The Polygon of forces is an extension of the Triangle of Forces. Pin a sheet of drawing-paper to the board.19. Fix three pulleys in any position on the board and hang weights on the cords as shown in Fig. 8 . pulleys. R and S are in equilibrium. Poly means many. Q. metal ring. This experiment could be regarded as ONE such joint on a structure. several weights. Polygon of Forces Method 1. Apparatus: Force board. so that the forces P. and it will be shown that in a system containing four or more forces. Student Objectives: The object of this experiment is to test that when four or more forces are in equilibrium at a point. they can be represented by a Polygon of Forces from which unknown forces can be found. 4. Put an arrow-head on each line the sense of the force and indicate beside each line the weight at the end of that cord. Conclusions 1. in magnitude. Represent the forces Q. Note the values of the forces. 7. and measure the line to scales. 6. Remove the paper and join the central mark to the other marks by straight lines. sense and direction? 2. How does the measured line compare with the observed value of P. S and R in order (i. Join the free ends of the lines representing Q and R. 9 .e. and one directly below each cord. State the theorem you set out to illustrate. Make a mark on the paper at the centre of the ring. 5. the arrows following each other) as shown in Fig. 20. EXPERIMENT NO. 3. Thus. friction force Ff is perpendicular to normal force N. 10 . Hence. 4 FRICTION . independent of contact area. 2. • Various wood blocks. dependent on the nature of the surface.is the frictional force resisting motion. Apparatus: • Inclined hardwood plane. and 4.CALCULATION OF COEFFICIENT OF FRICTION Objective: Determination of the coefficient of kinetic friction for pine on hardwood. either because particles are being broken off (wear). it takes some force to move bodies across each other when they are in contact. This is friction. Independent (within limits) of the relative speed between the surfaces. • mw . and for leather on hardwood. Therefore. or because the bodies are being separated slightly because of the projections on the surfaces. and • Balance. • Standard weights. for any given surface. and • m . Frictional resistance between surfaces is: 1. Theory: The smoothest solid surfaces are still uneven (microscopically or otherwise). Force Ff is parallel to the surfaces in contact and the normal force N is perpendicular to the surfaces in contact. the coefficient of kinetic friction is: where: • Ff . • Blocks with sand paper on the bottom. proportional to the force pressing the bodies together.is the mass of the wooden block and mass on top of it. • Blocks with leather on the bottom. • N .is the mass of the weight in kg attached on the string.is the normal force perpendicular to the surface. 5. Do several trials for loads of 0. 3. 11 . 7. and 400g weights on the block. 200. 6. Adjust the hanging weight (mw) so that the block (when started) will move at a CONSTANT SPEED (Very important). attach a weight (mw) to a block lying on the board with a string. 100. 2. Weigh the blocks that you will use to slide down the board and record the masses on the data table.Procedure: 1. and determine friction coefficient for each load. 4. 300. Repeat the procedure with a block that has sand paper on the bottom. With the plane horizontal and secured. Repeat the procedure with a block that has leather on the bottom. Place the string across the pulley so that the weight can hang freely from the end of the board. You should use Excel to create these graphs. Show values for µ on the vertical axis and values for m on the horizontal axis. 1. 5. Graph µ = f(m) for each case. Graph µ = f(mw) for each case. In addition to data collected and calculated (in the chart) answer next questions and create graphs. You should use Excel to create these graphs. Derive the formula µ = Ff/N = mw/m 4. 12 .Answer the following questions and create the graphs and include them on your lab report. Why is coefficient of friction constant or why not? 3. Show values for µ on the vertical axis and values for mw on the horizontal axis. Why is the pulley necessary? 2. the force acting down the plane just overcomes the friction and sliding takes place.5: FRICTION ON AN INCLINED PLANE Teaching Element: When a block is placed on an incline the tendency is for the block to slide down the plane. weights. but the force pressing the surfaces together decreases. pulley. At the Angle of Friction. Student Objectives: To investigate friction on the inclined plane and to investigate the relationship between the required forces (applied parallel to the plane) to slide a block up the plane. slider. If the angle of inclination is small the block is prevented from slipping by the friction between the surfaces. spring balance. cord.EXPERIMENT NO. the force exerted down the plane due to the weight of the block also increases. Friction on Inclined Plane 13 . As the angle is increased. Apparatus Friction plate. Place the slider and weights on the friction plane. Gradually pull the spring balance and note the force registered just as motion commences. and attach the spring balance as shown in the diagram. Keep the slider moving with uniform speed and again observe the reading on the spring balance. 3. 2. Conclusion Is it more difficult to start a body moving or to keep it moving? 14 .Method 1. EXPERIMENT NO.6: ANGLE OF FRICTION Teaching Element: When a block rests on a horizontal plane its WHOLE weight presses on the plane and the pressure between the surfaces sets up a resistance to movement which is called friction. The angle at which sliding begins to take place is called the ANGLE OF FRICTION and this experiment will show the relationship which exists between this angle and the Coefficient of Friction. Apparatus A plane as shown in sketch.4 15 . when the plane is inclined at an angle between the horizontal and vertical. sliders as used in Experiment No. As the angle of inclination increases. To show that the Coefficient of Friction is equal to Tangent of the Angle of Friction. the force acting along the plane increases but the force pressing the surfaces together decreases and so the friction force decreases. 2. Object To determine the angle of friction for various materials in contact. Student Objectives: 1. If the plane is vertical no pressure takes place between the surfaces because the whole weight is acting downwards parallel to the plane and the block will slide down the plane. and to find the connection between angle of friction and coefficient of friction. PART of the block weight acts parallel to the plane and PART of the weight produces pressure between the surfaces. Therefore. To measure the Angle of Friction and from it find the Coefficient of Friction. At a certain angle the force acting down the plane will overcome the frictional resistance between the surfaces and sliding will take place. 2.Friction on Inclined Plane Theory The ratio F/N mentioned in Experiment No.” and so find tan θ and θ (the angle of friction). 4. In the above diagram tan θ = H/L Method 1.4 is usually denoted by µ. Tilt the plane. Observations: 16 . Place a slider on the plane. until the slider just moves.” the corresponding length “L. Repeat for each slider. 3. Measure the height “H. Is tan θ = µ? 2. Now state the laws which you have obtained from each experiment on friction. Compare the values of tan θ with the values of µ for the same materials. Does the area of contact have any effect on the force of friction? 3.Conclusions: 1. 17 . Does the ratio F/N vary with different materials or is it a constant for all materials? 4. weights.EXPERIMENT NO. scale-pan. Friction is usually regarded as wasteful. Even so-called "smooth" surfaces have microscopic roughness which causes friction and the friction force must be overcome before sliding can take place. the minute. This proportion is called the COEFFICIENT OF FRICTION. Friction which opposes movement from rest is called STATIC FRICTION. The laws are only approximately true. In designing machines where sliding takes place. In practice it is found that the friction force is a fixed proportion of the force pressing the surfaces together. as in machines where it absorbs power and causes wear. This resistance to sliding is called friction.7: SLIDING FRICTION Teaching Element: When two rough surfaces are made to slide over one another. slider. Student Objectives: To verify the Laws of Friction and to measure the Coefficient of Friction for different materials. Friction opposes sliding and depends on the roughness of surfaces in contact. for example in friction brakes. As soon as sliding takes place it is found that less force is required and this is called KINETIC FRICTION. cord. the effect of friction must be taken into account and for this the LAWS OF FRICTION are used. uneven surface particles resist the sliding and are sometimes torn away. but it can be useful. 18 . but they form a useful and practical basis for dealing with friction problems. pulley. Apparatus: Friction plane. N lb (weight of slider and added weight. What conclusion may be drawn regarding the ratio F/N? 19 . Repeat the experiment for several loads. Place a weight on the slider. Note the reaction. Set up the apparatus as shown in above 2.) 3. Plot the F-N graph Observations: Conclusions: 1. F lb. Note the force. Add weights to the scale-pan..Friction on Inclined Plane Method: 1. Calculate the ratio F/N. needed to cause motion. 5. until motion commences. 4. Is the graph a straight line passing through the origin? 2. 6. Does the ratio F/N vary with different materials or is it a constant for all materials? 5. 20 . Now state the laws which you have obtained from each experiment on friction. What is the ratio F/N called? 4.3. This is called the PRINCIPLE OF MOMENTS. Student Objectives: The object of this experiment is to verify the Principle of Moments for parallel and nonparallel forces. The product is called the TURNING MOMENT of the force. If a body has several forces applied to it which have turning effects in opposite directions. Apparatus: Two spring balances. A method of calculating the effect of turning forces to produce equilibrium is to say The Moments Clockwise = The Moments Anti-Clockwise.EXPERIMENT NO 8: PRINCIPLE OF MOMENTS Teaching Element: When forces produce a turning effect. It can be applied both to parallel forces and to oblique forces. The Principle of Moments is frequently used in engineering and building work where forces have to be balanced to prevent any turning movement. the body will not turn if the total turning moments in each direction are equal. cord. 21 . but in all cases. several weights. when calculating the turning moment the length is the perpendicular distance from the pivot to the line of the force. this turning effect can be measured by the product of the force and the perpendicular distance between the pivot and the line of the force. Beam Forces Theory: Vertical component = AB sin ά Horizontal component = SB cos ά Method: 1. Arrange the apparatus as in Fig. Pull cord A. 4. Conclusion: Do the graphical and calculated and calculated values agree with the values found by experiment? 22 . making sure that the angle between A and C is 900. and the spring balance readings SA and SB. Find the horizontal and vertical components of SB graphically and by calculation. 3. Note the angle ά. 2. the weight W. the arms are of unequal length. In the slide balance. The pivot (or beam) balance makes use of this principle for weighing. the weight to be measured is placed in one pan and is balanced by known weights in the other. both pans being at the same distance from the pivot. Apparatus: Wooden beam. the weight being measured is placed in the pan on the short arm and balanced by a known weight which slides along the long arm. In the beam balance. Student Objectives: The object of this experiment is to demonstrate that the action of weighing with a beam balance or slide balance is based upon the Principle of Moments. A scale marked on the long arm is calibrated to show the weight in the pan. the body will not turn if the turning moments in each direction are equal. A turning moment being the force multiplied by the perpendicular distance from the centre of the pivot. two spring balances. 23 .EXPERIMENT NO 9: THE PIVOT (OR BEAM) BALANCE Teaching Element: Experiment No 5 shows that if a pivoted bar has forces applied to it which have turning effects. various small weights. Conclusions: 1. Place some weights on the beam.) 5. note the reading on the spring balances. What do you notice about the sum of the upward forces and the sum of the downward forces? 24 . 2. Respectively. 7. 6.) 3. 4. Before placing any loads on the beam. Calculate the reactions on the supports. Read the spring balances. Do the calculated values agree with the observed values? 2. Repeat the experiment for several different loadings. The differences (P-P1) and (Q-Q1) give the reactions on the supports due to the added weights. (These values are P and Q lb. (Let these be P1 and Q1 lb.Beam Forces Method: 1. Suspend the beam from the two spring balances as shown in Fig. 10 FORCES IN FRAME STRUCTURES 25 .EXPERIMENT NO. 26 . triangle or rectangle) cut from light colored cardboard. you will determine the center of gravity of some flat shapes and show that they will balance if supported at this point. with straight or curved sides.g.g. 11 Finding the point where an Object’s mass acts Topic Center of gravity and equilibrium Introduction The weight of a body is the force that the body exerts down towards the Earth. You will also investigate the stability of objects and show how an object will remain stable (in equilibrium) if its center of gravity is supported over its base. to which it is attracted by gravity. each having 4 – 5 holes punched around the edges (shapes can be regular or irregular..Experiment No. a large nail) knitting needle pencil 1 meter rule For Part B: 4 bottle corks (identical) 4 toothpicks 6 toothpicks cut in half to make 12 sticks of equal length and pointed at one end small board (about 20 ⋅ 30 cm) such as a cutting board 27 . and a longest dimension of about 15 – 20 cm) support stand and clamp 1 meter fine string or thick thread small weight (e. In this experiment. The center of gravity of a body is the fixed point through which all the weight of the body appears to act.. Time required Part A: about 10 minutes per shape 30 minutes for Part B Materials For Part A: number of shapes (e. Attempt to balance the shape on a finger placed at the point where the lines intersect (see diagram 4 on the next page). 8. 6. 5. Remove the plumb line and shape from the needle. 2. The clamp should be about 45 cm above the surface of the bench or table. Secure the needle in the clamp as shown in diagram 1 below. 4. (When suspended. Insert the needle through one of the holes in the shape and secure the plumb line around the needle as shown in diagram 2 above. a plumb line points directly towards the Earth’s center of gravity and thus shows the vertical line. Repeat steps 3 to 5 using all the holes on the cardboard shape. 28 . 7. Connect the dots showing the position of the plumb line to make a line (see diagram 3 below). Use the pencil to make a series of dots marking the line taken by the plumb line on the surface of the cardboard shape. Repeat steps 3 to 7 for each cardboard shape. Select one of the cardboard shapes. 3. Tie the small weight to one end of the fine string or thick thread to form a plumb line.Procedure Part A: Determining the center of gravity of a flat shape 1.) Make a loop at the other end of the string so that the plumb line is about 30 cm long. You will then have a series of lines as shown in diagram 3 below. To make object A.Part B: Equilibrium 1. The four sticks should enter the cork to a depth of about 5 mm and point directly downwards. carefully push four of the short sticks into one of the corks as shown in diagram 5A below. 29 . The four sticks should enter the cork to a depth of about 5 mm and splay out. carefully push four of the short sticks into the fourth cork as shown in diagram 5D above. 3. carefully push four of the short sticks into a third cork as shown in diagram 5C above.2. Position them in the cork as you did for object A. carefully push the four long sticks into another cork as shown in diagram 5B above. 30 . To make object D. 4. To make object C. To make object B. The four sticks should enter the cork to a depth of about 5 mm and angle in. Analysis Part A: Determining the center of gravity of a flat shape 1. Where would you estimate the center of gravity of the objects to be? 2. What happens if the shape is supported at this point? Part B: Equilibrium 1. How do you relate the position of the center of gravity to the order in which the objects toppled? 31 . Do the lines drawn on each cardboard shape meet at a single point? 2. to form a large “V” shape. direct ly under that balance po int. with the toothpick forming the center point. there is nothing over the base. The angle will be very different. This t ime. We can find that center of gravit y wit h some masking tape and a pen. in between the two forks.12 . we balanced by keeping the center of gravit y direct ly o ver the base. you will need: 1. I thought we would take the Science of Balance a bit further. wit h the apple at the point of the V. 2. 5. You want to st ick a strip of masking tape fro m one fork to the other. 32 . potato or other firm vegetable about 2 inches square a wooden toothpick or match st ick masking tape a marker or ink pen WStart by st icking a fork into one side of the piece of apple. it is more commo n than you might think. and try to balance the forks. Now you have a “W”. Now push the toothpick into the apple unt il only about half an inch st icks out. Place the po int of the toothpick on your finger. To try this. Again. M O R E S C I E N C E O F B A L A N C E For this week’s experiment.EXPERIMENT NO. While it may sound strange. Here. You want them both angled downward slight ly. you had to have the center of gravit y direct ly above the base. so that it passes directly under the po int where the toothpick is balanced on your finger. That mark shows you the center of gravit y for the object. Stick the toothpick into the apple. Then st ick the other fork into the other side. 2 forks a piece of apple. and the forks forming the two arms. We saw in the first experiment that for an object to balance. It works! The whole thing will balance quit e easily. we are going to reverse that to see that you can also balance an object by having the center of gravit y direct ly below its base. 3. Then make a mark on the tape. This is a case where an object’s center of gravit y is outside the object. but you will find that the mark you made on the tape is st ill direct ly under the balance point. Balance the forks on your finger again. but then there is nothing below the base eit her. How? In our previous experiment. balance it on your finger. point ing in t he same direct ion as the handles o f the forks. 4. If you can get it to balance. you will find that the mark is again underneath the balance po int. Take some t ime to play wit h this experiment. It is fun to see what you can balance it on. Then tell them the science behind it and impress them again. Use it to impress your friends. including a string stretched between two chairs. Have a wonder-filled week. if you have a steady hand. Try balancing it again. 33 .Remove the toothpick and st ick it into the apple in a different spot. Experiment 13 to verify the laws of limiting friction Take a block of wood of specific mass. Hence. the frictional force pulls it to the left. a thread. law three is verified. stone block and note the weight of the pan. Now add a few weights in the empty pan. Note the weight of the block + the weight on it and the weight on the pan. This verifies the fifth law.e. You will notice that they increase or decrease proportionally. Now consider any one of the blocks. a pan and a few weights and arrange them as shown in the figure. Hence. The block does not move. Change the face of the surface of contact and position of the block. This shows that even though the string pulls the block to the right. 1st law is verified. This observation will verify the law of static friction i. This verifies the law of limiting friction. 2nd law is verified. Keep adding weights in the pan and on the block so that the block just begins to move. pulley. it is tangential to the surface of contact. the fourth law. Hence. You will notice that the weight in the pan will be the same for all cases (when the block just begins to move). Since the frictional force f acts horizontal to the surface. 34 . Replace the wooden block with a glass block. Now add some additional weight on the block and adjust the weight on the pan so that the block just begins to move again. EXPERIMENT NO. 1 Procedure 35 .14 TO VERIFY LAMI’S THEOREM The simple arrangement used to verify the Lami’s theorem is generally called parallelogram law apparatus as shown in Fig. 36 . 15 TO VERIFY TRIANGLE LAW The simple arrangement used to verify the triangle law is generally called parallelogram law apparatus as shown in Fig.EXPERIMENT NO. 1 PROCEDURE 37 . 38 .