Purpose To study experimentally Hooke's Law, and to verify the relationship between the displacement of a spring (stretch or compression due to an applied force) and the restoring force exerted by the spring. Theory A spring hanging vertically from a support with no mass attached has a length Z, called its rest length. When a mass is added to the spring, and it is allowed to come to rest, the length of the spring is increased by x. The equilibrium position of the mass is now a distance _ + x from the spring's support. Since the attached mass is at rest, we conclude that the downward pull of gravity on the added mass is balanced by some upward force. This upward force is the spring's restoring force, Fearing = -KX, where x is the displacement vector of the spring from its rest length, and & is the force constant of the spring (also called the spring constant). The negative sign indicates that the direction of the restoring force is opposite to the displacement of the spring. The magnitude of the spring's restoring force can simply be stated as F = kx; at rest the force applied to the spring is equal in magnitude to the restoring force: At rest, ZF = 0, hence: Fiping = kx XC +x- my = kx F.= mg If a spring obeys Hooke's law, then a graph of applied force against extension will be a straight line, whose slope is k: Applied Force, F rise, AF Hope Extension, x The equation of the straight line is: F = kx where: F = stretching force applied to the spring k = spring constant x = extension of the springB anstretched pring 0900 It takes twice as much forces to stretch a FI spring twice as far. 2F A: There is no applied force, the spring is not stretched (except by under its own weight). B: A mass has been attached to the spring which initiates a gravitational force on the spring. stretching the spring by a distance x. C: Twice the mass as in B has been applied to the spring, doubling the force, and doubling the stretch, indicating a linear relationship. Materials Hooke's Law Simulation website https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html Procedure 1. Open the Hooke's Law simulation and click on the Lab section. 2. Click to add a check mark on the 'Displacement' and 'Moveable Line' options Displacement on the top right of the screen. Natural Length Mass Equilibrium Movable Lie Period Trace 3. Click and drag the 100 g mass onto the end of the spring. The spring will begin to oscillate up and down. Stop this by clicking on the red stop button at the top-middle of the screen. 190 9 4. Adjust the movable red line to the tip of the green arrow. 5. Click on the ruler tool on the right side of the screen and drag it over to the middle of the screen. Line up the end of the ruler with the blue natural length line so that you can measure the extension of the spring, represented by the green displacement arrow, at the red moveable line. 6. The ruler measures length in millimeters (mm). Measure the extension of the spring to the nearest tenth of a millimeter. A close-up of the ruler shows that the red moveable line is 20between the 16 mm and 17 mm marks on the ruler, closer to the 17 mm mark. To the nearest tenth of a mm the extension can be estimated as 16.8 mm or 0.0168 m. The reading you get may be slightly different so do not record 16.8 mm as your first measurement unless that is the value you observe when you follow these steps. 7. Record your first extension measurement for the 100 g mass in the Observations table. 8. Use the Mass slider at the top of the Hooke's Law simulation (just to the left of the spring support) to increase the mass by 40 -50 g. DO NOT CHANGE THE SPRING CONSTANT OR MOVE THE RULER. The mass will again oscillate a little; stop the motion with the red button at the top of the screen to the right of the spring support. Click on and drag the red moveable line to the tip of the green displacement arrow. Record in the Observations table the new mass ing and kg; calculate and record the force for the new mass; measure and record the extension of the spring in mm and m. 9. Repeat Step 8 for three additional mass values (for a total of 5 different masses). Observations Mass (g) Mass (kg) Force (N) Extension (mm) Extension (m) 100 g 0.100 kg 0.981 N Analysis 1. Return to the Hooke's Law simulation. Remove the known mass from the spring and replace it with the unknown red mass. Measure the extension of the spring with the red mass attached. Using the experimentally determined spring constant, calculate the value of the unknown red mass. Show the workings for the mass calculation.Workings Extension = m Red Mass = kg 2. Repeat the above step for the blue mass. Again, show the workings for the mass calculation. Workings Extension = m Red Mass = kg 3. Put the known mass back on the spring and set its value to 100 g. Click on Gravity the dropdown menu under Gravity on the right side of the simulation and What is the value of gravity? change this option to "Planet X". Measure the extension of the spring on Planet X and use this value (along with the experimentally determined spring Peretx constant) to find the value of g on Planet X. Show the workings for the calculation. Workings Extension = m g = m/s2 Question Mass is added to a vertically hanging rubber band and the extension is measured with the addition of each mass. The recorded data is displayed in the table below. Based on this data, does the rubber band obey Hooke's Law? Explain. Mass Added Extension, x (kg) (m 0.100 0.10 0.200 0.20 0.300 0.35 0.400 0.55 0.500 0.80