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Physics 141 Forces and Vector -- 1
FORCES AND VECTORS GOAL: To study the additions of vectors in the context of forces in equilibrium INTRODUCTION: If an object is not accelerating then the forces acting on the object must add up to zero. If this condition holds, one says the object is in equilibrium. Experimentally, the absence of acceleration is the "proof" that the sum of the forces is zero. Because people are better at seeing that an object is not moving than it is at judging constant motion, one can make this experimental test easier to perform by starting the system at rest, i.e., NOT MOVING. Then one applies something like: "A body at rest tends to stay at rest unless acted on by a nonzero net force."
and
(3) Much of this experiment, involves experimentally adding two forces ( and ) and measuring the force ( ) required to counteract this sum and keep the system at rest. Q i.
How is the force, , from a suspended ( nonaccelerating) object related to the mass, m, of the object?
(4) A force has both a magnitude and a direction, so forces must be added as vectors. In most cases one adds vectors by resolving each vector into perpendicular components ( say horizontal and vertical, or x and y), next add the components and then combine the components.
PROCEDURE: I Tug of war VS the TA. The class is divided in half and each group pulls on an end of a rope, This is basically a tug of war, but the group pull hard but establish an equilibrium. The TA then attempts to push the center of the rope sideways while the two groups of students try to prevent it. TENSION IN A VERTICAL PLANE
Figure 1 Vector addition and equilibrium. To add forces and (at angles " and $ respectively) that add together to get the resulting in force (with associated angle, (), one first does:
(1) Then
(2)
Figure 2 The pulley framework mounted on the counter.
Physics 141 Forces and Vector -- 2
There is a set of equipment for studying the vector nature of forces consisting of a framework on which two spring balances and a pulley are mounted as shown in Fig. 2. This setup will be used for parts II, II and IV. The position of the pulley can be adjusted up or down to change the configuration of the system, and the position of the suspended mass can be adjusted by moving the binder clip. The angles " and $ can be found by measuring Ax , Ay , Bx , and By.
THE FORCE TABLE
II. First, test the system without any suspended mass or mass hanger. Q ii.
What relationship do you expect between the readings of the two spring balances in this arrangement? Change the position of the pulley and test your hypothesis.
III. Forces in a Vertical Plane Try several (3) different masses on the system for each of the following three configurations of the suspension point for the mass: 1. Point below the bottom of the pulley. 2. Point above the bottom of the pulley. 3. Point at same level as the bottom of the pulley. Find each individual forces at the point where the mass is suspended ( at the binder clip) and compare to the expected values.
Figure 3
The remainder of this experiment will be done using a force table, see Fig. 3. This is an elevated metal disk about 40 cm in diameter. Angles in degrees are ruled near the rim of the disk and pulleys can be attached to the rim. Masses are suspended by strings that run over pulley and thus exerts a force on the center ring. If the center ring (while not touching the center pin) is stationary then the sum of these forces from the suspended masses is zero.
IV. A Different Set of Axes The obvious choice of axes, when working in a vertical plane is for the x-axis to be horizontal (left and right) and the y-axis to be vertical ( up and down). There are cases such as problems dealing with incline planes where this choice may complicate the mathematical operations rather than simplify them. Consider the following configuration on the framework. One string is about 30° from the horizontal and the other string is at a right angle to the first. Set this up. Use the corner of a piece of paper or a book as guide to setting the right angle. Experimentally this resolves the force associated with the suspended mass into two perpendicular components. Identify the new axes. Compare the experimental results with the calculated values for a few suspended masses. Relate this new choice of axis to the inclined plane problem, the normal force and the force down the plane.
The force table.
The magnitude of the force associated with a suspended mass is its weight ( NOT the length of the string). The direction of this force on the center ring is outward from the center toward the pulley and can be read from the protractor scale on the edge of the force table.
NOTE: In the following procedures, each force is written as an angle and an added mass. This mass must be added to the mass of the mass hanger before calculating the force.
1-D Equilibrium ( do very quickly) First, test the general idea that two equal but opposite forces cancel each other and leave the system V.
Physics 141 Forces and Vector -- 3
in equilibrium. Do this on the force table with only two pulleys and two strings. Place 200 grams on one of the mass holders, with the associated pulley at 0°. Q iii. How much mass do you expect you will need to put on the other mass holder to bring the system into equilibrium, and at what angle? Try it and test your hypothesis. This cancellation of forces, such that the system remains in equilibrium, is central to the rest of this experiment.
VI. Resolving a Force into Components Forces can be resolved experimentally into perpendicular components using a force table. First, set one pulley at 0° and a second pulley at 90°. Next, set the third pulley at the angle of the force given in the table below and add the required mass to the third mass hanger. Finally, balance the forces by adding the right amount of mass to the first and second mass hangers. Q iv. What would you need to do if the component of the force in the 0°-direction was negative?
Resolve the following forces ( added masses and angles): Mass #3 Angle grams #3 200 225° 200 200° 200 190° 200 164° Consider displaying your data in a table with enough columns to include various calculated values.
VII. Choice of Axis when Adding Forces Adding vectors at right angles is relatively simple because, with a clever choice of coordinates, the forces to be added are already resolved. Consider the similarities of adding the following pairs of forces (added pairs of masses):
Mass #1 grams 200 200 200 200
Angle #1 175° 30° 0° 351°
Mass #2 grams 50 50 50 50
Angle #2 265° 120° 90° 81°
If necessary set up on the force table for a few of these configurations to help visualize the systems. (This series of measurements is more about a concept than about numbers!)
VII. Adding Perpendicular Forces Consider adding two perpendicular forces, for example, one in the x direction and one in the y direction. Q v. How, in words, do you expect the angle of the resultant force to vary with the ratio of the xdirected force to y directed force? ( increase or decrease?) Q vi. And how mathematically does the angle depend on this ratio? Test these hypotheses, using all three pulleys and mass hangers. Experimentally determine the third force required to keep the system in equilibrium if the following pairs of forces ( added masses) are applied Mass #1 grams
Angle #1
Mass #2 grams
200 200 200 200 200 200
0° 0° 0° 0° 0° 0°
50 100 150 200 300 400
Angle #2 90° 90° 90° 90° 90° 90°
A i. Compare the force required to maintain equilibrium with the calculated sum of the two forces. (Consider displaying your data in a large table that includes various calculated values of interest including the ratio of the two applied forces) Is there any symmetry amongst these situations?
Physics 141 Forces and Vector -- 4
VIII. Adding Non-Perpendicular Forces Compared to adding perpendicular forces, adding two forces that are not perpendicular is more complex in terms of the mathematical operations that are required, but on the force table the procedure is identical. Q vii. As the angle between the two applied forces increases, what do you expect the counteracting force to vary? Test your hypothesis with the following pairs of forces (added masses): Mass #1 grams
Angle #1
Mass #2 grams
200 200 200 200 200 200 200 200
0° 0° 0° 0° 0° 0° 0° 0°
200 200 200 200 200 200 200 200
Angle #2 15° 30° 45° 60° 90° 120° 150° 180°
EQUIPMENT: Force table with 3 pulleys Frame with two spring balances and a pulley Mass set and hangers Binder clips Meterstick & ruler In room Incline Plane Demo Large Hemp Rope Scissors String and fishing line Levels EQUIPMENT HINTS: The pulley frame work: Check the zero of the spring balances. Ask for help if they are off. Check the calibration of the spring balances. To determine the angles in this system, measure various horizontal and vertical distances ( Ax , Ay, Bx and By in the figure) and use a bit of trigonometry.
Use the binder clips rather than knots when suspending the mass from the string. The horizontal force table: Level the tables Use fishing line Remember that the mass of the mass hangers must be included in the calculations. When the forces are nearly balanced, tapping the center ring may help in finding the true equilibrium of the system by overcoming the small friction in the system. Check that the strings moved on the center ring to positions that are appropriate for the angles to the pulleys. ANALYSIS: A ii. How accurate are your measurements? What is the major cause of your inaccuracy? A iii. For the force table system, draw free-body diagrams for the center ring and the three suspended masses. How are the various forces in these diagrams related? Is there something special about their relationship if the system is in equilibrium? How is the free-body diagram for the center ring different when the suspended mass balance and the system is in equilibrium? A iv. For the system consisting of the framework and spring scales and suspended masses, draw the freebody diagram for the suspended mass. A v. How does the concept of tension enter into the analysis of these experiments? A vi. Consider the following two methods of analyzing your results: 1. Vectorially adding the two applied forces and the measured counterbalancing force. Because the system is in equilibrium, one can compare this sum of three forces to zero. 2. Calculate the sum of the two applied forces and compare it to the measured counterbalancing force. How does one calculate the relative difference from an expected value of zero?
FORCES AND VECTORS GOAL: To study the additions of vectors in the context of forces in equilibrium INTRODUCTION: If an object is not accelerating then the forces acting on the object must add up to zero. If this condition holds, one says the object is in equilibrium. Experimentally, the absence of acceleration is the "proof" that the sum of the forces is zero. Because people are better at seeing that an object is not moving than it is at judging constant motion, one can make this experimental test easier to perform by starting the system at rest, i.e., NOT MOVING. Then one applies something like: "A body at rest tends to stay at rest unless acted on by a nonzero net force."
and
(3) Much of this experiment, involves experimentally adding two forces ( and ) and measuring the force ( ) required to counteract this sum and keep the system at rest. Q i.
How is the force, , from a suspended ( nonaccelerating) object related to the mass, m, of the object?
(4) A force has both a magnitude and a direction, so forces must be added as vectors. In most cases one adds vectors by resolving each vector into perpendicular components ( say horizontal and vertical, or x and y), next add the components and then combine the components.
PROCEDURE: I Tug of war VS the TA. The class is divided in half and each group pulls on an end of a rope, This is basically a tug of war, but the group pull hard but establish an equilibrium. The TA then attempts to push the center of the rope sideways while the two groups of students try to prevent it. TENSION IN A VERTICAL PLANE
Figure 1 Vector addition and equilibrium. To add forces and (at angles " and $ respectively) that add together to get the resulting in force (with associated angle, (), one first does:
(1) Then
(2)
Figure 2 The pulley framework mounted on the counter.
Physics 141 Forces and Vector -- 2
There is a set of equipment for studying the vector nature of forces consisting of a framework on which two spring balances and a pulley are mounted as shown in Fig. 2. This setup will be used for parts II, II and IV. The position of the pulley can be adjusted up or down to change the configuration of the system, and the position of the suspended mass can be adjusted by moving the binder clip. The angles " and $ can be found by measuring Ax , Ay , Bx , and By.
THE FORCE TABLE
II. First, test the system without any suspended mass or mass hanger. Q ii.
What relationship do you expect between the readings of the two spring balances in this arrangement? Change the position of the pulley and test your hypothesis.
III. Forces in a Vertical Plane Try several (3) different masses on the system for each of the following three configurations of the suspension point for the mass: 1. Point below the bottom of the pulley. 2. Point above the bottom of the pulley. 3. Point at same level as the bottom of the pulley. Find each individual forces at the point where the mass is suspended ( at the binder clip) and compare to the expected values.
Figure 3
The remainder of this experiment will be done using a force table, see Fig. 3. This is an elevated metal disk about 40 cm in diameter. Angles in degrees are ruled near the rim of the disk and pulleys can be attached to the rim. Masses are suspended by strings that run over pulley and thus exerts a force on the center ring. If the center ring (while not touching the center pin) is stationary then the sum of these forces from the suspended masses is zero.
IV. A Different Set of Axes The obvious choice of axes, when working in a vertical plane is for the x-axis to be horizontal (left and right) and the y-axis to be vertical ( up and down). There are cases such as problems dealing with incline planes where this choice may complicate the mathematical operations rather than simplify them. Consider the following configuration on the framework. One string is about 30° from the horizontal and the other string is at a right angle to the first. Set this up. Use the corner of a piece of paper or a book as guide to setting the right angle. Experimentally this resolves the force associated with the suspended mass into two perpendicular components. Identify the new axes. Compare the experimental results with the calculated values for a few suspended masses. Relate this new choice of axis to the inclined plane problem, the normal force and the force down the plane.
The force table.
The magnitude of the force associated with a suspended mass is its weight ( NOT the length of the string). The direction of this force on the center ring is outward from the center toward the pulley and can be read from the protractor scale on the edge of the force table.
NOTE: In the following procedures, each force is written as an angle and an added mass. This mass must be added to the mass of the mass hanger before calculating the force.
1-D Equilibrium ( do very quickly) First, test the general idea that two equal but opposite forces cancel each other and leave the system V.
Physics 141 Forces and Vector -- 3
in equilibrium. Do this on the force table with only two pulleys and two strings. Place 200 grams on one of the mass holders, with the associated pulley at 0°. Q iii. How much mass do you expect you will need to put on the other mass holder to bring the system into equilibrium, and at what angle? Try it and test your hypothesis. This cancellation of forces, such that the system remains in equilibrium, is central to the rest of this experiment.
VI. Resolving a Force into Components Forces can be resolved experimentally into perpendicular components using a force table. First, set one pulley at 0° and a second pulley at 90°. Next, set the third pulley at the angle of the force given in the table below and add the required mass to the third mass hanger. Finally, balance the forces by adding the right amount of mass to the first and second mass hangers. Q iv. What would you need to do if the component of the force in the 0°-direction was negative?
Resolve the following forces ( added masses and angles): Mass #3 Angle grams #3 200 225° 200 200° 200 190° 200 164° Consider displaying your data in a table with enough columns to include various calculated values.
VII. Choice of Axis when Adding Forces Adding vectors at right angles is relatively simple because, with a clever choice of coordinates, the forces to be added are already resolved. Consider the similarities of adding the following pairs of forces (added pairs of masses):
Mass #1 grams 200 200 200 200
Angle #1 175° 30° 0° 351°
Mass #2 grams 50 50 50 50
Angle #2 265° 120° 90° 81°
If necessary set up on the force table for a few of these configurations to help visualize the systems. (This series of measurements is more about a concept than about numbers!)
VII. Adding Perpendicular Forces Consider adding two perpendicular forces, for example, one in the x direction and one in the y direction. Q v. How, in words, do you expect the angle of the resultant force to vary with the ratio of the xdirected force to y directed force? ( increase or decrease?) Q vi. And how mathematically does the angle depend on this ratio? Test these hypotheses, using all three pulleys and mass hangers. Experimentally determine the third force required to keep the system in equilibrium if the following pairs of forces ( added masses) are applied Mass #1 grams
Angle #1
Mass #2 grams
200 200 200 200 200 200
0° 0° 0° 0° 0° 0°
50 100 150 200 300 400
Angle #2 90° 90° 90° 90° 90° 90°
A i. Compare the force required to maintain equilibrium with the calculated sum of the two forces. (Consider displaying your data in a large table that includes various calculated values of interest including the ratio of the two applied forces) Is there any symmetry amongst these situations?
Physics 141 Forces and Vector -- 4
VIII. Adding Non-Perpendicular Forces Compared to adding perpendicular forces, adding two forces that are not perpendicular is more complex in terms of the mathematical operations that are required, but on the force table the procedure is identical. Q vii. As the angle between the two applied forces increases, what do you expect the counteracting force to vary? Test your hypothesis with the following pairs of forces (added masses): Mass #1 grams
Angle #1
Mass #2 grams
200 200 200 200 200 200 200 200
0° 0° 0° 0° 0° 0° 0° 0°
200 200 200 200 200 200 200 200
Angle #2 15° 30° 45° 60° 90° 120° 150° 180°
EQUIPMENT: Force table with 3 pulleys Frame with two spring balances and a pulley Mass set and hangers Binder clips Meterstick & ruler In room Incline Plane Demo Large Hemp Rope Scissors String and fishing line Levels EQUIPMENT HINTS: The pulley frame work: Check the zero of the spring balances. Ask for help if they are off. Check the calibration of the spring balances. To determine the angles in this system, measure various horizontal and vertical distances ( Ax , Ay, Bx and By in the figure) and use a bit of trigonometry.
Use the binder clips rather than knots when suspending the mass from the string. The horizontal force table: Level the tables Use fishing line Remember that the mass of the mass hangers must be included in the calculations. When the forces are nearly balanced, tapping the center ring may help in finding the true equilibrium of the system by overcoming the small friction in the system. Check that the strings moved on the center ring to positions that are appropriate for the angles to the pulleys. ANALYSIS: A ii. How accurate are your measurements? What is the major cause of your inaccuracy? A iii. For the force table system, draw free-body diagrams for the center ring and the three suspended masses. How are the various forces in these diagrams related? Is there something special about their relationship if the system is in equilibrium? How is the free-body diagram for the center ring different when the suspended mass balance and the system is in equilibrium? A iv. For the system consisting of the framework and spring scales and suspended masses, draw the freebody diagram for the suspended mass. A v. How does the concept of tension enter into the analysis of these experiments? A vi. Consider the following two methods of analyzing your results: 1. Vectorially adding the two applied forces and the measured counterbalancing force. Because the system is in equilibrium, one can compare this sum of three forces to zero. 2. Calculate the sum of the two applied forces and compare it to the measured counterbalancing force. How does one calculate the relative difference from an expected value of zero?
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