Conservation of Mechanical Energy
Anupama Cemballi
Department of Physics, Case Western Reserve University
Cleveland, OH 44106-7079
Abstract:
Gravitation Potential Energy:
Introduction and Theory:
Gravitational potential energy is the potential energy an elevated object has due to the force of gravity. Potential energy is the stored energy an object has dependent upon its position. Near the surface of the Earth, the gravitational acceleration of an object is 9.8m/s^2. There is no uncertainty associated with this value since it has been measured countless times. The factors that affect the gravitational potential energy include m, mass, and the height of the object to an end point. The work done on an object due to the gravitational force is …show more content…
I weighed the cart and the mass by placing it on the balance; its mass was 997 grams. The uncertainty was estimated to be ±1 gram. The balance measures to the one’s place; therefore the most the measurement could be off is one gram. I tied a string to the cart and placed it over the wheel and attached the string to the counterweight, which included a weight hanger and weights. I then determined the hanging mass required to make cart move at constant velocity to determine the friction force on the system. I added paper clips to the counterweight and then pushed the cart towards the pulley, during which the photogate measured the speed. I continued to add paper clips until the speed measured was constant on Logger Pro, which recorded the speed of the cart as it was pulled along the track by the falling weight (the counterweight). The mass to balance the friction force was two grams (seven paperclips) with an uncertainty of 1.5 grams. The uncertainty is so large because the mass of the paper clips is so low, and the balance only measures to the one’s place. I then added an additional mass of about 30 grams. The Logger Pro program recorded the motion of the cart after I released it from …show more content…
The graph of energies versus distance (position) is attached. In the graph, the kinetic energy and the potential energy were calculated with grams instead of kilograms, so the values of slope and standard error must be divided by 1000. The best-fit line for total energy, E, is drawn on the graph. Its slope is -0.28263489 J/m with a standard error of 0.00208773 J/m. The uncertainty of ∆E/∆y can be calculated in quadrature using the following equation: δ ∆E/∆y=√((δ_(∆E/(∆y, slope))^2+δ_(∆E/(∆y, m))^2)) (6)
This value is 0.00208773 J/m, or 0.002 J/m, which is the same as the value calculated on Origin. So ∆E/∆y is -0.283±0.002J/m.
Spring Potential Energy:
Introduction and Theory:
Spring potential energy (aka elastic potential energy) is the stored potential energy of a stretched spring. Hooke’s Law states that the displacement of the spring when a force is acting upon it is directly proportional to that force that makes it compressed or extended. The equation below is Hooke’s Law,
F= -kx (7) where F is the force, k is the constant, x is the position. We can substitute in
F=mg (8) where m is mass and g is gravity, into Hooke’s Law, which gives us kx=mg (9)
Equation number (9) can be rewritten as
x=mg/k (10) which gives us the position as a function of the hanging mass. The the slope of the position versus mass graph is g/k,
Anupama Cemballi
Department of Physics, Case Western Reserve University
Cleveland, OH 44106-7079
Abstract:
Gravitation Potential Energy:
Introduction and Theory:
Gravitational potential energy is the potential energy an elevated object has due to the force of gravity. Potential energy is the stored energy an object has dependent upon its position. Near the surface of the Earth, the gravitational acceleration of an object is 9.8m/s^2. There is no uncertainty associated with this value since it has been measured countless times. The factors that affect the gravitational potential energy include m, mass, and the height of the object to an end point. The work done on an object due to the gravitational force is …show more content…
I weighed the cart and the mass by placing it on the balance; its mass was 997 grams. The uncertainty was estimated to be ±1 gram. The balance measures to the one’s place; therefore the most the measurement could be off is one gram. I tied a string to the cart and placed it over the wheel and attached the string to the counterweight, which included a weight hanger and weights. I then determined the hanging mass required to make cart move at constant velocity to determine the friction force on the system. I added paper clips to the counterweight and then pushed the cart towards the pulley, during which the photogate measured the speed. I continued to add paper clips until the speed measured was constant on Logger Pro, which recorded the speed of the cart as it was pulled along the track by the falling weight (the counterweight). The mass to balance the friction force was two grams (seven paperclips) with an uncertainty of 1.5 grams. The uncertainty is so large because the mass of the paper clips is so low, and the balance only measures to the one’s place. I then added an additional mass of about 30 grams. The Logger Pro program recorded the motion of the cart after I released it from …show more content…
The graph of energies versus distance (position) is attached. In the graph, the kinetic energy and the potential energy were calculated with grams instead of kilograms, so the values of slope and standard error must be divided by 1000. The best-fit line for total energy, E, is drawn on the graph. Its slope is -0.28263489 J/m with a standard error of 0.00208773 J/m. The uncertainty of ∆E/∆y can be calculated in quadrature using the following equation: δ ∆E/∆y=√((δ_(∆E/(∆y, slope))^2+δ_(∆E/(∆y, m))^2)) (6)
This value is 0.00208773 J/m, or 0.002 J/m, which is the same as the value calculated on Origin. So ∆E/∆y is -0.283±0.002J/m.
Spring Potential Energy:
Introduction and Theory:
Spring potential energy (aka elastic potential energy) is the stored potential energy of a stretched spring. Hooke’s Law states that the displacement of the spring when a force is acting upon it is directly proportional to that force that makes it compressed or extended. The equation below is Hooke’s Law,
F= -kx (7) where F is the force, k is the constant, x is the position. We can substitute in
F=mg (8) where m is mass and g is gravity, into Hooke’s Law, which gives us kx=mg (9)
Equation number (9) can be rewritten as
x=mg/k (10) which gives us the position as a function of the hanging mass. The the slope of the position versus mass graph is g/k,