Lab 4: Equilibrium & Lami s Theorem

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1 Dr. W. Pezzaglia Physics 1A Lab, Fall 2017 Page 1 Lab 4: Equilibrium & Lami s Theorem Preparation for Lab Read Young & Freeman section 5.1, and portions of chapter 4. ================================================================== Theory: A. Force 1. Newton s Second Law: defines force F as the cause of acceleration a with inertia m F ma (1) The SI units are Newtons (mass in kg, acceleration in m/s 2 ). 2. Gravity: exerts a force (called weight ) on any massive object, which follows the rule: F mg (2) where g is the gravity field of the earth. Near the surface of the earth it has a value of g=9.8 m/s 2 or 980 cm/s Weight Units: Today we will abuse the unit system and measure force in grams, because that s what our gauges read. But we should really multiply them all by g to get Newtons. So, for example, a 500 gram weight is really F mg ( 0.5kg)(9.8 m ) N (3) where g is the gravity field of the earth. Near the surface of the earth it has a value of g=9.8 m/s 2 or 980 cm/s 2. s 2017Sep13 All rights reserved W. Pezzaglia

2 Dr. W. Pezzaglia Physics 1A Lab, Fall 2017 Page 2 B. Equilibrium 1. Static Equilibrium: refers to a system which is at rest, i.e. just sitting there. By Galileo s law of inertia (restated as Newton s First Law), a system at rest will continue to stay at rest unless acted upon by a net nonzero force. In other words, the sum of the forces is zero (hence there is no net force, hence the acceleration is zero so the state remains unchanged). Stickman being pulled by two equal but opposite forces, hence will remain at rest, static equilibrium (until ripped apart) 2. Conditions for Equilibrium: Since force is a vector, we can state the condition for equilibrium as: The sum of forces along any direction must be zero. Or, we can state it as o Sum of forces in x direction must be zero o Sum of forces in y direction must be zero 3. Example: Block on Inclined Plane: W=mg Is the weight N Is the normal force f is the friction The vector sum must be zero, i.e. make a closed triangle W N f 0 (4) In component form the vectors would be, Weight W ( 0, mg) Normal N N( sin, cos ) Friction f f ( cos, sin ) (5a) (5b) (5c) The sum of the components in each direction must sum to zero: 0 F x f cos N sin (6a) 0 F y f sin N cos mg (6b) 2017Sep13 All rights reserved W. Pezzaglia

Dr. W. Pezzaglia Physics 1A Lab, Fall 2017 Page 3 C. Lami s Problem Bernard Lamy (1640-1715), book Traité de Mécanique (1679) showed the parallelogram addition of force vectors. 1. The Problem: Consider a weight suspended by 2 wires.

Hence we would have, F x F 2 cos60 F1 cos30 0 (7a) F y F2 sin 60 F1 sin30 mg 0 (7b) We have two unknowns {F 1, F 2 }, and two equations, so this can be solved (do you remember Cramer s Rule?). 3.

3 Dr. W. Pezzaglia Physics 1A Lab, Fall 2017 Page 3 C. Lami s Problem Bernard Lamy ( ), book Traité de Mécanique (1679) showed the parallelogram addition of force vectors. 1. The Problem: Consider a weight suspended by 2 wires. How is the weight shared by the wires? 2. Vector Addition by Components: The vector sum must be zero. Hence we would have, F x F 2 cos60 F1 cos30 0 (7a) F y F2 sin 60 F1 sin30 mg 0 (7b) We have two unknowns {F 1, F 2 }, and two equations, so this can be solved (do you remember Cramer s Rule?). 3. Lami s Solution: Based upon the law of sines (because the three vectors added together form a closed triangle). A B W (8) sin sin sin If we know the angles, then we can solve for A and B in terms of W. 2017Sep13 All rights reserved W. Pezzaglia

4 Dr. W. Pezzaglia Physics 1A Lab, Fall 2017 Page 4 Experiment: Part A: Setup Assemble setup, so that it is in equilibrium Weight: unlike the picture below, it would be better if you have the weight hanging on a string. The three strings should have a firm knot at the junction, without slipping. The force is calculated by the weight: W=mg. Force A on the left is measured directly by a spring gauge. This string is fixed. Force B on the right is delivered by a hung mass over a frictionless pulley. The force is calculated by F=mg. 2017Sep13 All rights reserved W. Pezzaglia

5 Dr. W. Pezzaglia Physics 1A Lab, Fall 2017 Page 5 Part B: Measurement When the system is stable (equilibrium), make your measurements Measure: Angles, the weights, and value on spring scale. DRAW: Scaled diagram, with correct angles, weights and spring scale Vector Diagram: Convert all weights to Newtons, and draw a scaled diagram of the force vectors. [i.e. angles must be correct, and the length of vector should be proportional to the force by some scale factor] Part C: Graphical Analysis Graphically Add the Vectors, by two methods, display diagram and answer questions: Question 1 Add your vectors by Head-to-Tail method Do they form a closed triangle as expected? Question 2 Add your vectors using parallelogram method (a) Add A+B, get resultant D (b) Then add D+W to get resultant R. Is R zero as expected? Question 3 Lami s Theorem Verify Lami s Theorem (see equation 8 above). Is it valid? Part D: Component Analysis Draw a coordinate system on your vector diagram (hint, put origin at the knot). While you can draw this coordinate system with any orientation, you will make your life easier if you make the y axis point upwards, so that vector W points directly along the negative y axis direction. Measure x and y coordinates of all vectors. Write them down in units of Newtons. a) Weight W (?,?) b) Left Force A (?,?) c) Right Force B (?,?) Question 4 Calculate the sum of the three vectors using component method. Show that the sum of the components in the x-direction is zero. Show that the sum of the components in the y-direction is zero. 2017Sep13 All rights reserved W. Pezzaglia

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