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Physics Laboratory 1 Last modified : 2007.4.2 Experiment 3. Worry of Sisyphus purpose of experiment The energy in nature is a conserved quantity. That is, the energy of an isolated system does not change as time varies. (It has time invariance) However, as the types of energy may vary, when we consider one type of energy it seems to disappear when the energy changes its type. The energy relevant to motion of a body under the earth’s gravitation is the gravitational potential energy and the kinetic energy, and the sum of these two is called mechanical energy and we call a system where the mechanical energy is conserved, a conservative system. When friction is involved in the motion of a body then the mechanical energy decreases as time passes because it can change its type into sound or heat due to friction. In this experiment, we investigate the transformation between the kinetic energy and the potential energy of a body in motion under the gravity of the earth and conservation of mechanical energy and the effect of friction. Experiment Outline Investigate the change of the kinetic energy as per motion of a body and that of potential energy as per the displacement under gravity. Measure each type of the mechanical energy and confirm that the sum is constant. (In this experiment, we investigate the ball rolling down a linear track with a PC camera. Plan on how to do the experiment in advance.) Investigate the difference when it rolls with and without slipping. The kinetic energy of a rolling body has both translational and rotational components. What is the expression for each component? What about the slipping case? What can you infer from this experiment? Draw a conclusion. Is the mechanical energy of the system you investigated conserved? If not, what is the reason? Can you quantify it? Experiment method The following devices are prepared for the experiment. (in parentheses are the number of the devices) A linear track (1) A circular track (shared by group 1 and 2) A ball (1) A ruler (1) A plastic dish to catch a ball (1) A linear(circular) track stand (2) A CCD (1) A balance (2, shared) If you need anything else, check with your teaching assistant or the experiment preparation room(19-111, 25-418) or prepare it yourself. You should study the balance and the micrometer in advance and make up a plan in advance. The following is the recommended experiment method. (movie) 1) Measure the time △t, as the function of height h, it takes the ball to pass the two points Z1 and Z2 and compare with the case in which the mechanical energy is conserved. ① set up the apparatuses as in the movie. Set up the camera so that the motion is observed from the side. Put the reference ruler on the same plane (where motion occurs) of the track. (*think about why we put the camera and the ruler there and discuss it in the result analysis. Turn on the computer and run “I-CA”. click [filecamera configuration] on the menu and make sure the CCD screen is turned on. (movie) * Tip. Camera configuration 1. After running I-CA program, select the camera configuration in the file menu. 2. Push the menu of the remote control till the setup menu pops up. 3. Select the ALS/AES on the third row. 4. Select the LEVEL -FIX- OFF on the last row. 5. Select 1/250 6. Push BACK on the remote control and return to the previous. 7. Select AGS/SENS 8. Select LIGHT and NORMAL 9. Select SENS 10. Select X32(32times) 11. Push BACK twice to exit ② put the bead at the height of h and start the experiment. Movie. When saving data is finished, select [picture-screen analysis] and analyze the saved data. Specify the datasaving path, determine the first and the last frame to analyze and the reference color of the object. Set up the reference point and ratio reference line and start the analysis. Movie movie think about what you should take for the reference line. (you can see the searching for the object in the designated analysis area from the movie) movie ③ after the analysis, save the data and you can see the image file used for analysis and the location data file of the object on the screen. (x and y coordinates as time flows) ④ you can obtain the coordinates of two points and the time △t it took for the bead to pass these two points from the data. Make a measurement more than five times for each height h and obtain the average and the stddev of each △t, and investigate for the height h of more than five different cases. ⑤ you can draw a graph obtained using Excel. (if you would like open data using Origin, open the data using Excel and select the file type as [text(delimited by tabs)] and save it and then open it with Origin) you can obtain the angle and the time △t of passing two intervals from the data. ⑥ Obtain the relation between the increase of the kinetic energy of the ball and the decrease of the potential energy from the measurement. (using two data (time) in a row from the data obtained by measurement you can obtain the average speed in the perpendicular direction at the certain interval, and using this average height of two points you can obtain the increase and decrease change of the kinetic and the potential energy. ⑦ What can you infer about the mechanical energy this system from this? Make sure you obtained the correct kinetic energy of the bead rolling on the track. 2) repeat the above experiment using beads of different masses. What difference can you see when the masses change? What is the reason? 3) find hmin the minimum starting height where the bead starts to fall without reaching the maximum of the circular track and compare with the theoretically expected value. Can you explain the difference, if any? If the bead is to overcome the maximum of the circular track without falling, what should be the speed of the bead at the maximum? Compare the result from the data and the theory. If there is a difference, think about why. 4)investigate the effect of friction from the rail track. Taking an appropriate height where the bead begins to roll, measure the height (from data) to which the ball returns and see the effect of friction that reduces the kinetic energy. background theory How do the bodies of two at a distance affect (gravitation) each other? Consider a gravitational field generated in the neighborhood of a body and suppose that this field applies a force on another body, by way of understanding the force between bodies at a distance. Since a body under a force moves with acceleration, its speed and kinetic energy increase. If an external force exists then the change of the mechanical energy may be understood as the work done by the external force, however, taking account of the energy of the external force source, the mechanical energy is still conserved. (i.e. total energy considering the energy of external force source is still conserved.) Acceleration of a body by the gravitational field is regarded as by the potential energy of the body under the gravitational field. That is, a potential energy means a power that can be transformed into a kinetic energy or that can do work. Quantitatively, the potential energy is defined as follows. To move a body of mass m with constant velocity under the gravity Fg = - mg (g is the gravitational acceleration) the external force (1) f = - Fg (2) that cancels this gravitational force should be applied. The work done by the external force to move this body from the displacement xo to x with constant velocity is (3) since the kinetic energy of a body with constant velocity does not change, this work done by the external force has to increase other type of mechanical energy. That is, it increases the potential energy and the change of the potential energy is (4) since the potential energy is defined by change as in the equation (4), its absolute value is undetermined and has no meaning. Therefore take an appropriate position of reference(eg. The surface of the earth) and set the potential energy at that position as some convenient value(eg. Uearth’s surface=0) then the gravitational potential energy at the height h of this body is U = mgh In this process of obtaining the potential energy, the relation, (5) E = T + U = 1/2 mv2 + mgh (6) that the sum of kinetic and potential energy, i.e., the fact that the body’s mechanical energy is constant is already used, therefore the conservation of mechanical energy holds. The motion of the steel ball cannot be explained sufficiently by the motion of the center. This is obvious because the ball can have a rotating motion without changing the location of its center. Generally, the kinetic energy of a rigid body like a steel ball is the sum of the translational kinetic energy of center of mass and the rotational kinetic energy about the axis through its center of mass. The kinetic energy of a ball of mass m, the radius r, the speed vcm of motion of center of mass and the angular velocity ω (rotating angle per second) about the axis through the center is (7) where I is the moment of inertia about the axis through the center of the ball. The moment of inertia could be thought of as the inertia of rotational motion, and for a homogeneous sphere, (8) When a ball rolls without slipping, (9) Therefore, the mechanical energy in the equation (6) in the case of a ball rolling without slipping, E = T + U = (7/10) mv2cm + mgx (10) Beware that the velocity v is the velocity of the ball on the slope and x is the height(perpendicular) of the ball. Let us investigate the time △t it took the ball at rest at the height h to start rolling on the slope and to pass h1 and h2. Since the mechanical energy of the ball at rest at the height h is E = mgh (11) the center of mass velocity (of the center parallel to the slope) of the ball when passing the height x is, if conservation of mechanical energy is assumed, (7/10)mv2cm + mgx = mgh therefore (12) (13) Let θ the angle of the slope then the velocity vcm is (14) (15) (16) and integrating (17) (18) i.e. (19) is obtained. Since this equation is obtained by the assumption of conservation of mechanical energy, by comparing the equation (19) with measurement of time t it took the ball at rest at the height h to start rolling on the slope and to pass h1 and h2, it can be shown whether the mechanical energy is conserved. In the case of the ball rolling not on a slope like this experiment but on a slanted track, the effective radius of rotation changes as (20) in the picture. d is the distance between the points where the ball touches the track. The equation (9) can be expressed using this effective radius (21) How, then, does the expression of (1) change? What about the case where the steel ball rolls with slipping? The minimum height for revolution of the circular track of radius Rcir can be calculated as follows. At the maximum of the circular track the total mechanical energy is (22) if the bead falls from the maximum of the circular track, even when there is no normal force of the circular track, the force of gravity on the bead at the maximum height when falling is the same as the centripetal force. (23) With this, you can obtain the minimum starting height hmin from where the bead can reach the maximum without falling ※ Questions The roller coaster must have a certain speed or more to move on the circular track at the maximum point. (this is obtained from the experiment) Also, at the end of the linear track (generally at the minimum height) it gives the maximum speed to double the fun. How high, then, the height should rise to double the speed at the minimum height? (ignore the attenuation from friction) when the speed is doubled, how much do the momentum and the kinetic energy increase? The law of energy conservation and translational symmetry in time of physical phenomena The law of energy conservation is one of the several conservation laws as the properties of nature that are important in physics. As the law of momentum conservation and the law of angular momentum conservation, the law of energy conservation states that the total amount of energy of an isolated system does not change as time flows, this means the energy itself cannot vanish or created even though the inner parts constituting an isolated system can exchange energy among them or that the type of energy of the isolated system can change. What does it mean that the energy of a system does not change as time flows? It means that the energy of the system and natural phenomena related to the energy stays the same whenever measurement is made. This property is called symmetry with respect to translations in time. The laws of conservation are attributed to the symmetrical properties the system or nature has, the law of momentum conservation is from the fact that the properties of a system do not change when observer’s coordinate is translated in space, and the law of angular momentum conservation is from the fact that the properties do not change when the direction of the coordinate is rotated. These laws of conservation coincide with the statement that there is no absolute time axis or spatial axis in nature. - Fundamentals of physics by Halliday & Resnick Without friction between a bob (a) hooked to a wire and a body (b) connected to a spring, mechanical energy is conserved and their motion continues forever. In this case, regardless of the time when observation of motion starts, the law of nature (law of motion) can be found. ※ friction and the law of energy conservation When friction acts on a moving body, the mechanical energy of the system is not conserved. That is, the mechanical energy of the system changes as time varies. Since friction is an interaction between the body and other systems(a desk or air, etc.), the body is not an isolated system so the law of (mechanical) energy conservation for an isolated system does not hold. However, as friction converts part of the mechanical energy to thermal energy, the total energy including the heat of the total system (body + desk, body + air, etc.) is conserved. This is the law of energy conservation in a broad sense. The exchange between the kinetic energy and the potential energy occurs freely but the transformation between the thermal and mechanical energy has a direction. That is, the mechanical energy is transformed by friction into the thermal energy, but the thermal energy is not transformed into the mechanical energy spontaneously. Why is that? This directional phenomenon in nature is called the second law of thermodynamics. Without this direction in transformation between the mechanical energy and the thermal energy, a huge cruise ship could sail by using the thermal energy for cooling the seawater without fuel. This property of nature called the second law of thermodynamics provides us with food for thought such as the impossibility of perpetual engine and time’s arrow flowing in one direction, etc. References A method of processing the measurement data analysis by graphs Return to Index