Physics Chapter 7: conservation of energy
What is the work done by normal force when an object slides along a surface?
0
Calorie
Amount of energy needed to raise temperature 1 gram of water 1 degree C
If conservation of energy is a law of nature, why do we have programs—like mileage requirements for cars or insulation standards for buildings—designed to encourage energy conservation?
Because some energy is transferred whether that be internally or through heat that we cannot harness in a usable way.
Where is elastic potential energy located?
It is located inside the spring
first law of thermodynamics
energy goes from the surrounding into the system and the energy goes out of the system to the surrounding
For which of the following systems is (1) mechanical energy conserved and (2) total energy conserved? (a) the system is isolated, and all forces among its constituents are conservative; (b) the system is not isolated, and work is done on it by external forces; (c) the system is isolated, and some forces among its constituents are not conservative
(1) (a) only; (2) (a) and (c)
Big Calorie
1000 cal 'food calorie' 1 Cal=4186J
Why can't the roller coaster reach D if the total energy is more than the PE at D?
Even though the car can't get to D, the total energy still exceeds the potential energy at D. But the car is blocked from reaching D by the potential barrier of peak C. We say that it's trapped in a potential well between its turning points.
Formula for when sliding friction is present
KE+PE+Ethermal =constant
mechanical energy
Kinetic or potential energy associated with the motion or position of an object
What does it mean if total energy exceeds the maximum potential energy for a roller coaster?
Therefore, it can move anywhere from its initial position at A. Since it's initially moving to the right, it will clear peaks B and C and will end up at D still moving to the right—and, since D is lower than A, it will be moving faster than it was at A
When the KE is a minimum (over this stretch of track), what is the direction of the acceleration?
down . When the KE is min, it is on the top of the round hill. For circular motion with constant speed, the acceleration is toward the center of the circle.
What does a potential energy graph for elastic look like?
looks like a parabola ( stretch =positive) (compressed = negative)
Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. 1) Rank in order from largest to smallest the speeds of the balls, v1, v2, and v3, just before each ball hits the ground.
v1 = v2 = v3
Work done on the system by the surroundings
positive
Formula for elastic potential energy
potential energy of a spring = 0.5 x spring constant x (extension)^2
What is work done by external force and work done by gravity when a mass is moved h distance at constant speed?
wext=+mgh Wgrav= -mgh This is always true if the kinetic energy at initial position is same as kinetic energy at final position
A projectile is fired with an initial speed vo at an angle from the horizontal. What is the KE of the projectile when it is on the way down at a height h above the ground? (Assume no air resistance.)
(1/2)mvo 2 - mgh
Consider Earth and its atmosphere as a system. Which of the following processes conserves the total energy of this system? (a) a volcano erupts, spewing hot gases and particulate matter high into the atmosphere; (b) a small asteroid plunges into Earth's atmosphere, heating and exploding high over the planet; (c) over geologic time, two continents collide, and the one that is subducted under the other heats up and undergoes melting; (d) a solar flare delivers high-energy particles to Earth's upper atmosphere, lighting the atmosphere with colorful auroras; (e) a hurricane revs up its winds, extracting energy from water vapor evaporated from warm tropical seas; (f) coal burns in numerous power plants, and uranium fissions in nuclear reactors, with both processes sending electrical energy into the world's power grids and dumping warmed water into the environment
(a), (c), (e), (f)
Gravitational force actually decreases with height, but that decrease is negligible near Earth's surface. To account for the decrease, would the exact value for the potential-energy change associated with a height change h be (a) greater than, (b) less than, or (c) equal to mgh, where g is the gravitational acceleration at Earth's surface?
(b) The potential-energy change will be slightly less because at greater heights, the gravitational force is lower and so, therefore, is the work done in traversing a given distance.
Suppose it takes the same amount of work to push a trunk straight across a rough floor as it does to lift a weight the same distance straight upward. If both trunk and weight are moved instead on identically shaped curved paths between the same two points as before, is the work (a) still the same for both, (b) greater for the weight, or (c) greater for the trunk?
(c) On the curved paths, the work is greater for the trunk. The gravitational force is conservative, so the work is independent of path. But the frictional force isn't conservative, and the longer path means more work needs to be done.
A mass slides down a rough ramp (with friction) of height h. Its initial speed is zero. Its final speed at the bottom of the ramp is v. While the mass is descending , its KE________ . While the mass is descending, its PE ______ While the mass is descending, its (KE + PE) = total mechanical energy _________
-KE increases -PE decreases -and the mechanical energy decreases because some energy is being converted into thermal energy
At peaks and valleys of the roller coaster the force is _______, what does this mean
0, force is the slope of potential energy curve
A block initially at rest is allowed to slide down a frictionless ramp and attains a speed v at the bottom. To achieve a speed 2v at the bottom, how many times higher must the new ramp be?
4 By conservation of energy, mgh = (1/2) m v2 , so h = v2 / (2g). If v is doubled, v2 increases by a factor of 4, and so does h.
A hockey puck sliding on an ice rink is moving at 1 m/s when it slides onto a carpet that someone left on the ice. The puck comes to rest after moving 1m on the carpet. How far along the carpet would the puck go, if its initial speed was 2m/s?
4 m
How much more KE does the puck have moving at 2 m/s compared to moving at 1 m/s?
4 times as much
Little calorie
4.186 J
A spring-loaded dart gun shoots a dart straight up into the air, and the dart reaches a maximum height of h. The same dart is shot straight up a second time from the same gun, but this time the spring is compressed twice as much before firing. How far up does the dart go this time, neglecting friction and assuming an ideal spring?
4h
If a book is being lifted a height h, why is it wrong to say that the potential energy is in the book?
Because the potential energy comes from the system of (book +earth +gravitational pull)
What is the work done by the two forces together in moving the object a distance x to the right as shown in the diagram? The magnitude of each force is F and the directions of the two forces are 90o apart.
Fx (The vertical force does no work at all.)
A spring-loaded gun fires a dart at an unknown angle above the horizontal. No air resistance. The spring constant k and the initial compression x of the spring are known. Can you solve for the maximum height h above the starting point using Conservation of Energy
No. The KE at the top of the trajectory is unknown, so you can't solve for h in terms of known quantities.
Consider two identical objects released from rest high above the surface of the earth (neglect air resistance for this question).In Case 1 we release an object from a height above the surface of the earth equal to 1 earth radius, and we measure its kinetic energy just before it hits the earth to be K1.In Case 2 we release an object from a height above the surface of the earth equal to 2 earth radii, and we measure its kinetic energy just before it hits the earth to be K2. 1) Compare the kinetic energy of the two objects just before they hit the surface of the earth.
K2 = (4/3) K1
A small mass, starting at rest, slides without friction down a rail to a loop-deloop as shown. The maximum height of the loop is the same as the initial height of the mass. NOTICE: there is no force holding the ball against the rail, except gravity. Will the ball make it to the top of the loop?
No, the ball will not make it to the top. The ball needs to have significant speed and significant KE at the top of the loop in order to remain in contact with the rail. If, instead of an open rail, the ball was in a tube, as in the question before, then the ball would make it to the top. The condition for maintaining contact with the rail at the top of the rail is normal force N = 0.
A hockey puck slides without friction along a frozen lake toward an ice ramp and plateau as shown. The speed of the puck is 4m/s and the height of the plateau is 1m. Will the puck make it all the way up the ramp?
No. The initial KE is (1/2)mv2 = 8m (in SI units). The PE at the top is mgh = 9.8m (in SI units). There is not enough initial KE to convert into PE.
Elevator 1 can carry a load of mass m up a distance h in a time t, before the engine overheats. Its power output is P1. Elevator 2 can carry the same load up twice the distance (2h) in twice the time (2t). What is its power output P2 ? Elevator 3 can carry twice the load (2m) up twice the distance (2h) in the same time (t) as elevator 1. What is its power output P3 ?
P2 = P1 P = deltaW/deltat = mgh/t P3 = 4P1
Suppose an unusual (non-Hooke's Law) spring has a force which is a constant, independent of positive stretch length x : Fspring = -C , where C is a positive constant. (Instead of Fsping = - kx). The potential energy associated with this spring is (for positive stretch x) is
PE C| x| Remember the definition of PE: deltaPEF = -WF = +Wext
Gravitational force is constant, how does potential energy increase?
Potential energy increases linearly with height
If the force is zero at a given point, must the potential energy also be zero at that point? Give an example.
Potential energy is kind of arbitrary, only the gradient matters. For example, a stationary battery has a load of chemical potential energy and yet there is zero force on it. Only changes in potential energy require forces to act
gravitational potential energy
Potential energy that depends on the height of an object. No work or potential-energy change occurs on the horizontal segments since the gravitational force is perpendicular to the motion.
A "system" consists of a mass m hanging from a spring (spring constant k) and the Earth as shown. In situation A, the mass is hanging freely, at rest, so the spring is stretched beyond its relaxed length. In situation B, the mass is also stationary, but the spring is now at its relaxed length, because an alien from Planet X has lifted the mass and is holding it up. In which situation is the total energy of the mass/spring/earth system bigger?
The answer is B. The alien did positive work on the mass/spring/earth system when he (it?) lifted the mass, so some energy left the body of the alien (he burned up some alien calories) and entered the mass/sping/earth system. If you add up the PEgrav = mgh and the PEelastic = (1/2)kx2 , you will also find that there is more energy in the mass/spring/earth system when the alien is holding the mass up.
potential-energy curve
The potential energy curve shows how much potential energy the system has at each position.
How does compression of spring affect the blocks distance?
The stiffer the spring or the more it's compressed, the farther the block goes. The greater the friction or the normal force mg, the sooner the block stops. If μ=0, mechanical energy is once again conserved; then our result shows that the block would slide forever.
What does it mean when the potential energy at the highest peak of the roller coaster exceeds the total energy?
Therefore, the car will move rightward from A, clearing peak B, but will come to a stop just before peak C, a so-called turning point where potential energy equals the total energy. Then it will roll back down to the left, again clearing peak B and climbing to another turning point where the potential-energy curve and total-energy line again intersect. Absent friction, it will run back and forth between the two turning points.
Is the conservation-of-mechanical-energy principle related to Newton's laws, or is it an entirely separate physical principle?
They are related because the forces are equal and opposite. From Newton's Third Law, the change in energy of a closed system can be derived.
Student A climbs a flight of stairs 10 m high in 8 s. Student B climbs a flight 20 m high in 16 s. Which student is producing the greater mechanical power output?
Tie. The mechanical power output is the power that is going into increasing the gravitational PE = mgh. Power = mgh/t
A spring is compressed a distance d by a block of mass m which is initially at rest on the floor. The kinetic coefficient of friction between the floor and the block is μk. The mass is released and the spring launches it to the right, where it eventually stops a distance D from its starting point. What is the macroscopic work done on the block by friction during this process?
Wfriction = -μkmgD The macroscopic work done by friction on the block is equal to the magnitude of the frictional force acting on the block times the displacment of the block. The sign of this work is negative since the force of friction is in the opposite direction to the displacement.
elastic potential energy
When you stretch or compress a spring or other elastic object, you do work against the spring force, and that work ends up stored as elastic potential energy
Can potential energy be negative? Can kinetic energy? Can total mechanical energy? Explain.
Yes potential energy can be negative. But kinetic energy can't however the change in kinetic energy can be negative .Total mechanical energy can be negative if you are taking energy out of the system.
A small ball, starting at rest, slides without friction along a frictionless tube in the shape of a loop. The maximum height of the tube is the same as the initial height of the ball. ASSUME NO FRICTION. Will the ball make it to the top of the loop?
Yes, the ball will make it to the top, by conservation of energy.
A car of mass m moving with speed v suddenly slams on the brakes and skids to a stop. Suppose I ask "What was the work done on the car by the force of friction?" Do you have enough information to answer the question?
Yes. Use the Work-KE theorem. The work done by the force of friction = the work done by the net force = delta KE.
How is the amount of thermal energy = -work done by friction
because E thermal = -W(non-conservative force) and since friction is a non conservative force change in E thermal = - Wfriction
Why can't we define a potential energy associated with friction?
because friction is dependent on the path taken
There is a climber and a mover that do work, but only the climber can recover the work done why?
because if the climber falls then the gravitation energy gives back the work done through kinetic energy but the frictional force doesn't give back the energy received from the mover.
Why is force of friction not a conservative force?
because the work done depends on the path taken, the longer the path the more work is done by friction
How do you find the change in potential energy of a force if the force is a conservative force?
change in potential energy of that force=-Wf=+Wext
two types of potential energy
gravity=mgh elastic=1/2(kxsquared)
conservative force
force involving no friction, work done depends only on the initial and final position and is independent of the path taken. It gives back the energy that was transferred by doing work.
thermal energy
heat energy; when temperatures increase, molecules vibrate, expand, and rise; this causes a variety of scientific phenomena studied in fifth grade
Conservation of Mechanical Energy
if there is no friction or air resistance, the mechanical energy of a system changes only if the system does work on something else or energy is added from outside the system. Emech=KE+PE
Potential-energy curve for two hydrogen atoms?
if total energy is less than zero then the two atoms bound, but if the total energy is more than zero then the two atoms are arbitrarily far apart and can't bond
A mass m is at the end of light (massless) rod of length R, the other end of which has a frictionless pivot so the rod can swing in a vertical plane. The rod is initially horizontal and the mass is pushed down with an initial speed vo . What initial kinetic energy is required for the mass to pivot 270o to the vertical position?
mgR You get the same answer regardless of where you set the zero of height.
Work done by the system on the surroundings
negative
If the difference in potential energy between two points is zero, does that necessarily mean that an object moving between those points experiences no force?
no because the force would be equal and opposite at that point
When the KE is max, what is the direction of the acceleration?
no direction, the acceleration is zero . When the KE is max, it is in the flatbottomed valley, moving with constant velocity.
A bowling ball is tied to the end of a long rope and suspended from the ceiling. A student stands at one side of the room and holds the ball to her nose, then releases it from rest. Should she duck as it swings back? Explain.
no, because Mechanical energy is conserved, so if the ball is released from rest, it cannot climb higher than its initial height.
If the potential energy is zero at a given point, must the force also be zero at that point? Give an example.
no, because the force can be moving the object through kinetic energy
Does static friction do work?
static friction create 0 thermal energy so it has 0 net work
Potential Energy (PE)
stored energy that can transform into other forms of energy like kinetic, elastic
Where the potential energy is changing rapidly, the force is ___________
the greatest
What happens to mechanical energy if non conservative forces act on the system?
the mechanical energy isn't conserved and gets transformed into internal energy
ΔK+ΔU+ΔEint=Wext Describe this formula?
where Wext is the work done on the system by forces acting from outside. If Wext is positive, then this external work adds energy to the system; if it's negative, then the system does work on its surroundings, and its total energy decreases.
what does W+Q=delta U mean?
work done +head added= change in energy of the system
A box sliding on a horizontal frictionless surface runs into a fixed spring, compressing it a distance x1 from its relaxed position while momentarily coming to rest. 1) If the initial speed of the box were doubled, how far x2 would the spring compress?
x2 = 2 x1
I move the zero of potential energy up to the starting point of the skateboarder (skateboarder still starts from rest). The total energy Etot of the system is now:
zero
The change ΔUAB in potential energy associated with a conservative force is the negative of the work done by that force as it acts over any path from point A to point B:
−∫B to A F⋅dr
