A platform of mass 0.83 kg is supported on four springs (only two springs are shown in the picture, but there are four altogether). A chunk of modeling clay of mass 0.65 kg is held above the table and dropped so that it hits the table with a speed of v = 0.91 m/s.
The clay sticks to the table so that the the table and clay oscillate up and down together. Finally, the table comes to rest 7.1 cm below its original position.
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a) What is the effective spring constant of all four springs taken together?
k = N/m
HELP: What is the net displacement Äx of the spring from its former equilibrium position? What force F was required to compress the springs? The effective spring constant is the ratio F/Äx.
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b) With what amplitude does the platform oscillate immediately after the clay hits the platform?
A = cm
HELP: You might find it useful to find the speed of the table+clay system just after the collision, using momentum conservation.
HELP: Look at this from an energy point of view. Just after the collision, the total energy is partly potential (since the spring is now stretched from its new equilibrium position) and partly kinetic. When the spring is at its maximum displacement from equilibrium, all the energy is potential.
HELP: Remember that for a vertical spring, one can ignore gravity if one takes the equilibrium position as the origin. The equilibrium position is the place where the restoring force of the springs equals the weight of the clay.
HELP: Just after the collision, what is the potential energy stored in the spring? Remember that it is now displaced from its new equilbrium position. What is the kinetic energy? The sum of these is the total energy.
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a) To find the effective spring constant of all four springs taken together, we need to calculate the net displacement of the spring and the force required to compress the springs.
The net displacement, Δx, of the spring from its former equilibrium position is given as 7.1 cm below its original position. However, we need to convert this to meters, so Δx = 0.071 m.
The force, F, required to compress the springs can be calculated using Newton's second law, F = ma, where m is the mass of the clay and a is the acceleration due to gravity.
First, we need to find the weight of the clay, W = mg. The mass of the clay is 0.65 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. So, W = 0.65 kg * 9.8 m/s^2.
Now, we can calculate the force, F = W, required to compress the springs. F = 0.65 kg * 9.8 m/s^2.
The effective spring constant, k, is the ratio of the force, F, to the net displacement, Δx. So, k = F/Δx.
We can now substitute the values to find the effective spring constant of all four springs.
b) To find the amplitude with which the platform oscillates immediately after the clay hits the platform, we can approach this problem from an energy point of view.
Just after the collision, the total energy is partly potential (since the spring is now stretched from its new equilibrium position) and partly kinetic. We can use the principle of conservation of energy to find the amplitude.
First, let's consider the potential energy stored in the spring just after the collision. When the spring is at its maximum displacement from equilibrium, all the energy is potential.
The potential energy stored in the spring can be calculated using the formula PE = 0.5 * k * x^2, where k is the effective spring constant calculated in part a), and x is the amplitude.
Next, let's consider the kinetic energy of the table+clay system just after the collision. We can use the principle of conservation of momentum to find the speed of the table+clay system.
Since the clay sticks to the table, the momentum before the collision is equal to the momentum after the collision. The momentum before the collision can be calculated as p = m * v, where m is the mass of the clay and v is the speed at which it hits the table.
The momentum after the collision can be calculated as p = (m + M) * V, where M is the mass of the platform and V is the speed of the table+clay system just after the collision.
From the conservation of momentum, we can equate the two expressions for momentum before and after the collision, and solve for V.
Now, the kinetic energy of the table+clay system just after the collision can be calculated as KE = 0.5 * (m + M) * V^2.
The total energy just after the collision is the sum of the potential energy stored in the spring and the kinetic energy of the table+clay system.
Setting the total energy equal to the potential energy, we can solve for the amplitude, A, using the equation: PE = KE.
Substitute the values into the equation and solve for the amplitude A.
Note: Make sure to take into account any unit conversions and use appropriate units while calculating the values.