Conservation of energy 2023 lab (1)

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Mechanical Engineering

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Dec 6, 2023

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Johanna Gantier Partner: Simrun Karin Instructor: Professor King Leung Lab: PHYS 11000- 1L12 March 24, 2023 Laboratory #5: Conservation of energy Objectives: The objectives of this lab are to study when mechanical energy is conserved and when it is not Equipment: Ultrasonic motion detector, cart, track, friction block, electronic balance, meter stick Introduction In this experiment we are going to consider two kinds of energy, kinetic energy and gravitational potential energy , and their sum, which gives as result the total mechanical energy. These concepts will be applied to a cart moving down an inclined track. The cart will start from rest (V=0), and then accelerate down the track. As it does, it speeds up, and its height above the table decreases, however, its acceleration increases. This means that its kinetic energy increases and its potential energy decreases. We want to see if these changes compensate for each other so that their sum balance and stay the same. The second part of the experiment is the same as the first, except that a friction block will be attached to the cart. We will again want to see if the mechanical energy is constant. From class you should remember that if an object of mass m is moving with a velocity v , then its kinetic energy is given by KE= (1/2) mv2 . An object of mass m, which is a height y above the origin, has a gravitational potential energy given by mgy , where g=9.80 m/s2 . If there is no other source of potential energy around (like a spring, for instance), then the total mechanical energy of the mass is just the sum of the two, E= (1/2) mv^2+mgy . If there is no friction present, this quantity will be conserved. If there is friction present, the mechanical energy will decrease, and we will have that. (Work done by friction) = (final mechanical energy)- (initial mechanical energy). The work done by the friction force, F f , is just – F f s , where s is the distance, the object moved. Because the work done by the friction force is negative , the above equation implies that the final mechanical energy is less than the initial mechanical energy. Procedure: 2. Measure the length of the track, l, the elevation of the higher end, h, and the mass of the cart, M.

Measure Results conversion Length of the track (L) (m) 161.5 cm 1.615m Elevation of the higher end (H)(m) 13.8 cm 0.138m Mass of the cart (kg) 529.8 g 0.5298kg  5.19N Mass of the friction block (kg) 121.6g 0.162kg  1.223N Angle 85.1 degrees KE 0.743J PE 0.717J ME 1.46J Acceleration of the cart (calculated) 0.148m/s 2 Acceleration of the cart (given by the graph) 0.147 m/s 2 Time (t) given by the graph 2.0s Calculations Length of the track, l = 161.5 × 1 m 100 cm = ¿  1.615m Elevation of the higher end, h = 13.8cm × 1 m 100 cm = ¿  0.138m Mass of the cart, M = 529.8g × 1 kg 1000 g = 0.5298 kg  F = ma→ ( 0.5298 kg ) ∗ ( 9.8 m s 2 ) = 5.19 N Mass of the friction block= 121.6g × 1 kg 1000 g = 0.162 kg  F = ma→ ( 0.162 kg ) ( 9.8 m s 2 ) 1.19 N 3. Go to the Data menu and then to the Modify Column feature. You will need to modify the equation for both the potential and the kinetic energy. In the equation for the kinetic energy replace 1 by the mass of the cart, which you measured. In the equation for the potential energy, replace the 1 by M(h/l). KE  replaced 1 by the mass of the cart = 0.5298kg PE= replaced 1 by M (h/l) = 0.036m M= Mass h= difference of the higher end and the lower end H-h L= length M (h/l) = ( 0.5298 kg ) ∗ ( 0.138 m − 0.028 m 1.615 m ) = ¿ 0.036m

4. You are going to hold the cart part of the way up the ramp and then let it go. Sketch your predictions for the kinetic energy versus time plot, the potential energy versus time plot, and the total mechanical energy versus time plot. Now do a run and sketch your results. Predictions: We predict that as explained in the introduction, that as the cart moves down the hill, the kinetic energy would increase while the potential energy will decrease. As a result of this the graph for KE will be an increasing line, the graph for PE will be a decreasing line and the mechanical energy will balance out. It will be shown as a constant line. Giving us the notion that although KE increases and PE decreases there is no net loss of energy, and the energy will be conserved. Results

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This graph represents the actual experiment in the lab. As we can see the kinetic energy (in green) increase the potential energy (in red) decreases. While mechanical energy should represent a constant line. However, due to some technical difficulties with the equipment cannot be properly represented on this graph. 5A. Derive an equation for the kinetic energy of a cart going down an inclined ramp in terms of h, l, g, and M. List the equation for the kinetic energy of a cart going down an inclined ramp in terms of h, l, g, and M. Kinetic energy equation  KE= 1/2mv 2 1-V= V o +at  V= at 2-ΣF=ma  mgSin = ma  a=gSin = (9.80 m/s 2 * 0.138m/1.615m) = 0.148 m/s 2 3-Substitute acceleration in equation #1 V= (gSin )*(t) Sin = H/L Derived Kinetic energy equation. KE= 1/2m(gSin t) 2  KE= 1/2m (gh/Lt) 2 Kinetic energy KE= ½(0.5298kg) (9.8m/s 2 *0.138m/1.615m*2.0s) 2 = 0.743 J

Potential energy PE=mgh PE= 0.5298kg*9.80 m/s 2 *0.138m= 0.717J Mechanical energy ME= KE+PE ME= 0.717J + 0.743J= 1.46J 5B. Open the file Kinetic Energy; you should see two sets of axes, one for a velocity versus time plot and another for a kinetic energy versus time plot. Do another experimental run, i.e., hold the cart part way up the ramp and let it go. You can use the Curve Fit feature (fit it with a quadratic function) from the Analyze menu to fit the kinetic energy versus time plot. Compare this to the result you derived. Equation given by the graph. Y=A+BX+CX 2  is a quadratic equation

A= 0.266 B=-0.302 C= 0.147 Y= 0.266 + (-0.302x) + 0.147x 2 C= the acceleration of the moving object B= is the final velocity of the moving object A= is the initial velocity of the moving object Kinetic energy can be transformed into other forms of energy and is transferred between objects. To accelerate an object, we need to apply force and to apply force we need to do work. When work is done, the energy is transferred to the other object, and it starts moving. On the run we examine the Kinetic energy slope which is represented as a parabolic quadratic function. The motion of an object in constant acceleration is modeled by x = ½ at2 + v0t + x0, where x is the position, a is the acceleration, t is time, and v0 is the initial velocity (NOTE: 1/2a = A; v0 = B; x0 = C on the graph equation: At^2 +Bt +C). This is a quadratic equation whose graph is a parabola. If the cart moved with constant acceleration while it was rolling, your graph of position vs. time will be parabolic. The Trail # 1 example shown fits a quadratic equation to the data. Based on this equation ½ at2 + v0t + x0, the acceleration equation is multiplied by 0.5. In the graph the given acceleration is 0.147 m/s 2 which is accurate with the acceleration of the object obtained in this first part 0.148 m/s 2 . 6. Now attach the friction block to the cart. You are again going to hold the cart part of the way up the ramp and then let it go. Sketch your predictions for the kinetic energy versus time plot, the potential energy versus time plot, and the total mechanical energy versus time plot. Now do a run and sketch your results. Prediction: We predict that that as the cart moves down the hill, the kinetic energy would increase while the potential energy will decrease. It will be shown as a parabolic curve since energy is lost due to friction. Giving us the notion that although KE increases and PE decreases. However, due to the friction force the energy will not be conserved.

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Results Measures Result Conversion Mass of the cart (kg) 529.8g 0.5298kg  5.19N Mass of the friction block (kg) 121.6g 0.162kg  1.223N Angle 85.1 degrees Ff 0.0011N Coefficient of friction µ 0.00143 Acceleration of the cart (calculated) 0.014m/s 2 Time (t) given by the graph 5.0s 7. Use the fact that the change in the mechanical energy is equal to the work done by friction to find the value of the friction force acting on the cart. This force is due to the friction block. Use the electronic balance to find the mass of the friction block, and then find the coefficient of friction between the friction block and the track. Mass of the friction block= 121.6g *1kg/1000g = 0.1216kg

Mass of the cart+ friction block= 0.1216kg+0.5298kg= 0.6514kg Wf= Ff*D Wf= Ff* (h/ sin θ ) Ff=M c g cos *µ Coefficient of friction ΣF=ma Ff= µF N F N = mg µmg=ma  µ= a/g µ= a/g  for acceleration we are using the acceleration given by the graph µ=0.014 m/s 2 ÷ 9.80 m/s 2 = 0.001432857= 0.00143 Friction force Ff=M c g cos *µ Ff= (0.5298kg) *(9.80 m/s 2 ) * ( Cos 85.1 o ) *(0.00143) Ff= 0.001097- 0.0011N conclusión: The law of conservation of energy states that energy can never be created or destroyed, only converted between types, KE and PE. As the roller coaster cart gets dragged to the top of the track its potential energy increases. Potential energy is stored energy, and the roller coaster has a particular kind called gravitational potential energy, or stored energy due to height. When the cart crests the hill, gravity pulls the cart down the hill and that potential energy is converted to kinetic energy or energy due to movement. The higher the track goes, the greater gravitational potential energy the roller coaster will have, and thus the more kinetic energy it will have as it rolls down the track. This is why taller roller coasters go faster. As we saw in this experiment, to study energy conversions such as the roller coaster, scientists can use energy graphs, where the amount of each type of energy is graphed over time for a particular object. Time is the independent variable, so it always goes on the x, or horizontal axis on a graph. The y axis is the dependent variable, or what we measure. In our case, it will be the amount of energy measured in Joules (J). Since we are talking about energy conversions, there will always be more than one type of energy on the graph, so we're going to need a key. Most energy graphs will have potential and kinetic energy. At any given point, an object on the graph will have the same amount of total energy unless an external force is applied as happened when we applied the force of friction (energy was not conserved). However, that total energy will be distributed in different amounts of kinetic and potential energy. With the help of the kinematic equations and the method of substitution we found the variable needed to find the KE, PE, and ME values as well as the values of Acceleration (which was accurate with the acceleration given by the graph), the friction coefficient and friction force acting upon the friction block, the track, and the cart. This was a very complicated but fun experiment. I hope my partner and I achieved the goal of this experiment.