Energy in Atwood s Machine

Transcription

1 Energy in Atwood s Machine In this lab you will use an energy analysis to analyze the motion of Atwood s Machine. Atwood s Machine (see illustration below) consists of two masses connected by a string stranded over a pulley. The heavier mass will fall to the ground, while the lighter mass will be pulled upward. Realistically, some friction in the pulley or other resistive force (such as air resistance) will act against the motion of the system. We can easily calculate the gravitational forces acting on the system, but there is no simple way to accurately determine the magnitude of the frictional forces. In this lab, you will use an energy analysis to do this. Principles The mechanical energy of a system is the sum of its kinetic (K) and potential energies (U): E = K + U Kinetic Energy is energy that an object has by virtue of its mass m and its speed v. It is defined as: Kinetic energy (and any form of energy) is measured in joules: 1 joule = 1 newton-meter. For a compound system, the total kinetic energy is simply the sum of the kinetic energies of the masses of which it is composed. Potential energy is energy an object has by virtue of its position in a force field. For a mass m located a vertical distance y above some reference point y 0, the gravitational potential energy is U = mg(y y 0 ) This is usually expressed as U = mgh, where it is implied that h = y y 0. It is important to recognize that potential energy is always relative some reference position (y 0 ), and that we are always free to set the potential energy at that reference position equal to zero.

2 The gravitational potential energy of Atwood s Machine is not simply the sum of the potential energies of its two masses. Instead, to find the potential energy of the system, we must consider the work done by the force of gravity as the system moves from the reference position y 0 to its final position y. The potential energy at position y is then the negative of the work done by gravity. Work: Roughly speaking, work is force applied over a distance: Work = force x distance. However, we must consider the direction in which the force acts, so we use the component of the force in the direction of displacement, d: F Work = Fdcosϕ ϕ d If the work done by the force is positive, the object s kinetic energy would be increased. If the work done is negative, then the object s kinetic energy will be decreased. If the work done is zero, then the force would not change the objects kinetic energy. This would be the case if the force F in the above diagram were perpendicular to the displacement (Φ = 90 degrees). Resistive forces like friction always act opposite to the motion of the object, so the angle ϕ is always 180 degrees and the work done by these nonconservative forces is always negative. For an ideal, frictionless, system, we would expect that its energy would not change as it travels. For the ideal Atwood s Machine, if we release it from rest from some vertical distance above the floor, we could easily calculate its final velocity from Total energy (K + U) at the final position = Total Energy (K + U) at the starting position.

3 However, non-conservative forces always decrease the total energy of the system. In this case we write the energy principle in the following form: The change in the mechanical energy of a system = The work done on the system by nonconservative forces. Or: E Final = E Initial + W NC Thus if we can measure the change in the total energy as the Atwood s Machine moves, we can determine the work done by resistive forces, and from that the magnitude of these forces.

4 1: Atwood s Machine Atwood s Machine is a simple elevator. Two masses, M 1 and M 2 are connected by a string and hung from two pulleys. If one mass is greater than the other, then the system will accelerate in the direction of the heavier mass. If the difference between the two masses is small, then the acceleration will be much less than the acceleration of gravity. It is straightforward to find the acceleration of the system using Newton s Laws. If we call the heavier mass M 1 and the lighter mass M 2, it is the difference in the weights of these masses that accelerates the system, and their combined mass is its inertia, so that: a = (M 1 M 2 ) g (M 1 + M 2 ) In practice there are always resistive forces acting, such as friction in the pulleys or air resistance against the masses, so that the predictions of the above expression may not be adequate. We need to include these resistive forces in the analysis, but there are no simple force laws for predicting their magnitude. Instead, we must determine this experimentally.

5 Experiment 1 Objectives Analyze the potential and kinetic energy of Atwood s Machine Determine how much energy is lost to friction during a displacement of the system. Predict the time of descent Procedures M 1 d M 2 1. Set-up the Atwood s Machine. Mount the double-pulleys on an upright rod using a right-angle clamp. Hang mass hangers from each end of a string strand this over the pulleys. Use enough string so that the descending mass will travel about 125 cm to the floor. Adjust the pulleys so that they are vertical and parallel with each other. Put 500 grams on each mass hanger. In the diagram above, M 1 represents the mass of the down-going hanger and includes all mass that is on the hanger. M 2 represents the total mass of the up-going hanger and its load. Use this notation in your analysis below. d represents the displacement of the system after it is released from rest and travels to the floor. You will want to represent this as Δy = y y 0.

6 2. Test Run. Put 10 extra grams on the down-going hanger and time its descent to the floor, starting from rest. Take some practice runs until you can measure consistent times, then average the time of descent over several runs. Safety note: Be alert for falling masses, and for wildly swinging mass hangers if one or both 500-gram masses come off their hangers. This can happen when one mass hits the floor at high speed. Use kinematics to calculate the expected time of descent for a frictionless system and compare this to your measured time. Measure the distance of descent for M 1 to the nearest millimeter. (Note: the string will stretch a bit over time, so re-measure the distance during the experiment.) Record your data, calculations and conclusions. 2. Develop your energy analysis: How does the energy of the system depend on the masses and their displacement? You probably found that leaving friction out of the analysis did not give a good prediction for the time of descent. To find the force of friction acting on the system you can use the principle that the change in mechanical energy of a system is equal to the work done on it by non-conservative forces: ΔE = W NC Diagram the system in your lab notebook, and indicate with arrows all the forces acting on the system weights, tensions, and the presumed frictional force. Also indicate a coordinate axis for measuring displacements of the system. Find an expression for the total kinetic energy of the system by adding the kinetic energies of the individual masses. Find an expression for the potential energy of the system when it is located at an arbitrary position y above the floor (let y 0 = 0).

7 The potential energy of the system at position y is the negative of the work done by gravity if the system were to move from y=0 to y. Alternatively, it is the work you would have to do to move it at constant speed to position y. Considering that the two weights act against each other, write down an expression for the work you would have to do to displace the system to an arbitrary height above the floor. Do not consider friction here. Note that the tension forces in the string do no net work on the system: explain why. Add your expressions for the kinetic and potential energies of the system to find an expression for the mechanical energy of the system. This expression should be in terms of M 1, M 2, the velocity of the system at any time, and its displacement above the floor. Confirm your expressions with you instructor, as they will be your working theory. 3. Find the energy loss and the force of friction in the system. Using your expression for the mechanical energy of the system, calculate its change in energy as the system move from its initial position (where v = 0) to its final position (where y = 0). To do this, you will need to know the system s final velocity. Use kinematics and your previously measured time to find this. This change in energy must be due to the work done on the system by the force of friction while it descends. Since this work is just the force of friction time the distance the system traveled, you can find the force of friction acting on the system. You can assume that this force is constant. 4. Find the actual acceleration of the system and recalculate the theoretical time of descent. Now that you know the magnitude of the frictional force on the system, revise the original expression for its acceleration. Use: a = F net /M total. Recalculate the expected time of descent using your revised value for the acceleration. Compare this with your measured time. Record your calculations and results.

8 5. Run the system with 15 and 20 grams added to the down-going hanger. Use the procedures above to find the friction force acting in these cases. In each case, first predict the time of descent. You can use the previous value for the force of friction on the system to predict its acceleration. Find the energy loss and the force of friction as above, and revise your prediction for the time of descent, if necessary. In your analysis, determine if the force of friction changes significantly when the mass of the system changes. Safety Note again: For 20 grams accelerating mass in particular, catch the system as the down-going mass hits the floor. When one of the weights fall off its hanger, the hanger can be slung around violently.