Apply the following warm-up to the side-by-side configuration, and then repeat for the end-to-end configuration: 1. Make a picture of the spring configuration similar to each of the drawings in the Equipment section (Figure A and Figure B). Draw a coordinate system. Label the positions of each unstretched spring, the final stretched position of each spring, the two spring constants, and the mass of the object suspended. Put arrows on your picture to represent any forces on the object. Assume that the springs are massless. For the side-by-side configuration, assume that the light bar attached to the springs remains horizontal (i.e. it does not twist). For each two spring configurations make a second picture of a single (massless) spring with spring constant k' that has the same object suspended from it and the same total stretch as the combined springs. Be sure to label this picture in the same manner as the first. 2. Draw force diagrams of both spring systems and the equivalent single spring system. Label the forces. For the end-to-end configuration, draw an additional force diagram of a point at the connection of the two springs. 3. Apply Newton's laws to the object suspended from the combined springs and the object suspended from the single replacement spring. Consider carefully which forces and displacements will be equal to each other. For the end-to-end configuration: Draw an additional force diagram for the connection point between the springs. At the connection point, what is the force of the top spring on the bottom spring? What is the force of the bottom spring on the top spring? 4. Solve your equations for the effective spring constant (k') for the single replacement spring in terms of the two spring constants.
Apply the following warm-up to the side-by-side configuration, and then repeat for the end-to-end configuration:
1. Make a picture of the spring configuration similar to each of the drawings in the Equipment section (Figure A and Figure B). Draw a coordinate system. Label the positions of each unstretched spring, the final stretched position of each spring, the two spring constants, and the mass of the object suspended. Put arrows on your picture to represent any forces on the object. Assume that the springs are massless.
For the side-by-side configuration, assume that the light bar attached to the springs remains horizontal (i.e. it does not twist).
For each two spring configurations make a second picture of a single (massless) spring with spring constant k' that has the same object suspended from it and the same total stretch as the combined springs. Be sure to label this picture in the same manner as the first.
2. Draw force diagrams of both spring systems and the equivalent single spring system. Label the forces. For the end-to-end configuration, draw an additional force diagram of a point at the connection of the two springs.
3. Apply Newton's laws to the object suspended from the combined springs and the object suspended from the single replacement spring. Consider carefully which forces and displacements will be equal to each other.
For the end-to-end configuration: Draw an additional force diagram for the connection point between the springs. At the connection point, what is the force of the top spring on the bottom spring? What is the force of the bottom spring on the top spring?
4. Solve your equations for the effective spring constant (k') for the single replacement spring in terms of the two spring constants.
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