Springtime in Paris
Marie and Pierre hope to be students at Ecole Polytechnique, the most prestigious engineering school in France. They currently attend the Lycee Henri IV (a secondary school). Their science teacher has provided them with a pair of springs, a wheeled cart, and a set of labeled masses with which to do some experiments designed to improve their understanding of the physics of springs. The springs each have very little mass.
They begin their experiments by hanging one of the springs from a horizontal rod, and then attaching a mass of 100.0 grams to the end of the spring. Pierre uses his hand to slowly lower the attached mass until he can detect that, if he removes his hand, the attached mass will not accelerate either upwards or downwards. After Pierre has removed his hand, Marie measures a stretch of 13.0 cm for the spring as it supports the hanging mass.
Part A. What is the spring constant for the spring which is hanging from the horizontal rod?
Next, using the same method as for the 100.0-gram mass, they hang first a 200.0-gram mass and then a 300.0-gram mass from the end of the spring. They find that the spring stretches, respectively, twice and then three times as much as in the case of the 100.0-gram mass. They are now convinced that the spring is relatively ideal. Similar tests on the other spring convince them that the second spring is essentially identical to the first. They end their first set of experiments by hanging their wheeled cart from the end of the vertical spring. As a result, they find that the spring stretches 80.6 cm .
Part B. What is the mass of the cart?
For a second set of experiments, Marie and Pierre trap the cart between the two springs, oriented horizontally East-West. In the laundry room at Pierre's house, the walls are a little less than 2.0 meters apart and the floor is very smooth and level linoleum. From the local hardware store, they get two hooks that can be screwed into a wooden wall. They put one hook into the baseboard of the western wall of the laundry room, and the other hook directly across from the first into the baseboard of the eastern wall. They attach one spring to the westernmost hook, attach the eastern end of that spring to the western end of the cart, and then attach the other spring between the eastern end of the cart and the easternmost hook. Each spring is always stretched tightly enough so that it never touches the ground (each spring may always be considered to be perfectly horizontal at all times). Since each spring is always stretched at all times, the force on the cart by the western spring is always Westward, and the force on the cart by the eastern spring is always Eastward. In its equilibrium position, the cart is exactly halfway between the two walls.
Marie and Pierre begin by doing some calculations designed to predict the effective spring constant of their setup. First they consider the horizontal forces acting on the cart if it is displaced 1.0 cm to the East of its equilibrium position.
Part C. When the cart is displaced by 1.0 cm to the East of its equilibrium position, what is the net force on the cart by the two springs?
Give the East-West component of the net force, with East positive.
Good work. Marie and Pierre also calculate that if they displace the cart by 2.0 cm to the East of its equilibrium position, then the net force on the cart by the two springs will be 0.302 N to the West, or exactly twice the net force which results from the 1.0-cm displacement.
Part D. As a result of their experiments, Marie and Pierre decide that, if they choose to call the East-West direction the x direction, and if they choose to assign the equilibrium position of the cart as x = 0, then writing Newton's Second Law for the cart results in a simple relationship. The relationship determined by Marie and Pierre is given below, but one number with units has been left blank; you must fill in that missing value. Here is the relationship:
Fill in the blank in the above relationship with the proper value.
Marie and Pierre know that, if the cart is set into vibrational motion by pulling it to the East and releasing it from rest, they should be able to predict many properties of the motion from this one relationship (from Part D). They decide to perform a series of experiments in which they vary the initial displacement and the total mass of the cart. To vary the total cart mass, they tape selected labeled masses to the top of the cart; the tape is used to keep the masses from slipping as the cart accelerates during its vibrational motion. Marie and Pierre decide to use four different combinations of cart mass and initial displacement, and they make a table with the expected values of the time for ten complete vibrations, the total energy of the cart and springs during the vibrations (assumed to be constant in each case), and the maximum cart speed for each of the four combinations.
In Parts E-H, you are asked to fill out the entries in the table compiled by Marie and Pierre. You will be given a cart mass and an initial displacement size. In each case, you must predict the expected values of the time for ten complete vibrations, the total energy of the cart and springs, and the maximum cart speed for that particular combination.
Part E. Cart mass of 670.0 grams and an initial displacement size of 5.0 cm :
Give the time for ten complete vibrations, followed by a comma, followed by the total energy of the cart and spring, followed by a comma, followed by the maximum speed of the cart.
Part F. Cart mass of 670.0 grams and an initial displacement size of 10.0 cm :
Give the time for ten complete vibrations, followed by a comma, followed by the total energy of the cart and spring, followed by a comma, followed by the maximum speed of the cart.
Part G. Cart mass of 1340.0 grams and an initial displacement size of 5.0 cm :
Give the time for ten complete vibrations, followed by a comma, followed by the total energy of the cart and spring, followed by a comma, followed by the maximum speed of the cart
Part H. Cart mass of 1340.0 grams and an initial displacement size of 10.0 cm :
Give the time for ten complete vibrations, followed by a comma, followed by the total energy of the cart and spring, followed by a comma, followed by the maximum speed of the cart.
Part I. What is the maximum acceleration experienced by the cart at any time during any of the four experiments included in Marie and Pierre's table?
Pierre Does an Experiment Designed to Determine ?s Between the Cart and a Mass Riding on the Cart
As a final experiment, Pierre decides to use their experimental set-up to measure, in a way, the coefficient of static friction between the cart and a mass which is riding on top of the cart without being taped in place. He places a 170.0-gram mass on top of the cart, then grabs the cart with his hand, displaces the cart to the East by a measured amount, and finally releases the cart to see if the mass slips when the cart starts into motion. He repeats this procedure many times, each time increasing the amount of displacement, until he finds that the mass does slip. He finds that, for any displacement greater than 25.0 cm , the mass always slips when the cart is released.
Part J. What is the coefficient of static friction between the cart and the mass riding on the cart?