course Phy 231 6/20 6:30 011. `query 11*********************************************
.............................................
Given Solution: `a** A conservative force conserves energy--you can get your energy back. Push something massive up a hill, then climb back down the hill. The object, by virtue of its position, has the potential to return most of your energy to you, after regaining it as it rolls back down. You will have done work against gravity as you move along a path up the hill, and gravity can return the energy as it follows its path back down the hill. In this sense gravity conserves energy, and we call it a conservative force. However, there is some friction involved--you do extra work against friction, which doesn't come back to you. And some of the energy returned by gravity also gets lost to friction as the object rolls back down the hill. This energy isn't conserved--it's nonconservative. ** Another more rigorous definition of a conservative force is that a force is conservative if the work done to get from one point to another independent of the path taken between those two points. Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qIf a system does work W1 against a nonconservative force while conservative forces do work W2 on the system, what are the change in the KE and PE of the system? Explain your reasoning from a commonsense point of view, and include a simple example involving a rubber band, a weight, an incline and friction. Your solution: Change in KE is equal to the nonconservative work +the conservative work Change in PE is the opposite of the conservative work KE is going to be like moving a car faster than it was going before. The amount of movement is determined by the forces acting on it. Pulling a car up an incline with a hanging weight, the conservative force is gravity pulling on the weight and causing the car to move forward. The nonconservative force is friction; that is negative. It takes away some force and the change in KE is not as great. PE is increased if the conservative force is decreased (gravitational force is lowered, or the weights move up instead of down) and is decreased if the weights fall. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** `dKE is equal to the NET work done ON the system. The KE of a system changes by an amount equal to the net work done on a system. If work W1 is done BY the system against a nonconservative force then work -W1 is done ON the system by that force. `dPE is the work done BY the system AGAINST conservative forces, and so is the negative of the work done ON the system BY nonconservative forces. In the present case W2 stands for the work done on the system by conservative forces, so `dPE = - W2. PE decreases, thereby tending to increase KE. So work -W1 is done ON the system by nonconservative forces and work W2 is done ON the system by a conservative force. The NET work done ON the system is therefore `dW_net_on = -W1 + W2. The KE of the system therefore changes by `dKE = -W1 + W2. If the nonconservative force is friction and the conservative force is gravity, then since the system must do positive work against friction, W1 must be positive and hence the -W1 contribution to `dKE tends to decrease the KE. e.g., if the system does 50 J of work against friction, then there is 50 J less KE increase than if there was no friction. If the work done by the conservative force on the system is positive, e.g., gravity acting on an object which is falling downward, then since force and displacement in the same direction implies positive work, gravity does positive work and the tendency will be to increase the KE of the system and W2 would be positive. A couple of numerical examples: If W2 is 150 J and W1 is 50 J, then in terms of the above example of a falling object, this would mean that gravity tends to increase the KE by 150 J but friction dissipates 50 J of that energy, so the change in KE will be only 100 J. This is consistent with `dW_net_ON = -W1 + W2 = -50 J + 150 J = 100 J. The previous example was of a falling object. If the object was rising (e.g., a ball having been thrown upward but not yet at its highest point), displacement and gravitational force would be in opposite directions, and the work done by gravity would be negative. In this case W2 might be, say, -150 J. Then `dKE would be -150 J - 50 J = -200 J. The object would lose 200 J of KE. This would of course only be possible if it had at least 200 J of KE to lose. For example, in order to lose 200 J of KE, the ball thrown upward would have to be moving upward fast enough that it has 200 J of KE. If a force and the displacement are in opposite directions, then the work done by that force is negative. If the system moves in a direction opposite the force exerted BY the system, the work done BY the system is negative. Note, however, that if this is the case then any equal and opposite force exerted ON the system will be in the direction of motion, so the force will do positive work ON the system. A separate document related to this problem is located in the document work_on_vs_by_dKE_dPE_etc_questions_answers.htm &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qIf the KE of an object changes by `dKE while the total nonconservative force does work `dW_nc on the object, by how much does the PE of the object change? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The PE change is going to be equal to the work done by the nonconservative force minus the change in KE. dWnc = dKE+dPE. DKE and dPE have to add up to the nonconservative force. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** We have `dKE + `dPE + `dWbyNoncons = 0: The total of KE change of the system, PE change of the system and work done by the system against nonconservative forces is zero. Regarding the object at the system, if W_nc is the work done ON the object by nonconservative forces then work -W_nc is done BY the object against nonconservative forces, and therefore `dWnoncons_on = -W_nc. We therefore have `dKE + `dPE - W_nc = 0 so that `dPE = -`dKE + W_nc. ** Equivalently, the work-energy theorem can be stated `dW_ON_nc = `dKE + `dPE In this example the work done on the system by nonconservative forces is labeled W_nc, without the subscript ON and without the `d in front. However it means the same thing, so the above becomes W_nc = `dKE + `dPE and we solve for `dPE to get `dPE = -`dKE + W_nc &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* OK ********************************************* Question: `qGive a specific example of such a process. Your solution: On object is being pulled by a car. The total nonconservative work done on the object is -50J. The change in kinetic energy is 10 J. You put this in the equation dPE = -50J 10J = -60J This means that the potential energy changed by -60J or 60J is removed from the potential energy. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** For example suppose I lift an object weighing 50 N and in the process the total nonconservative force (my force and friction) does +300 J of work on the object while its KE changes by +200 J. The 300 J of work done by my force and friction is used to increase the KE by 200 J, leaving 100 J to be accounted for. More formally, `dW_noncons_ON = +300 J and `dKE = +200 J. Since `dW_noncons_ON = `dKE + `dPE, So +300 J = +200 J + `dPE, and it follows that `dPE = +100 J. This 100 J goes into the PE of the object. ** There is no specified connection between the 300 J of work I do and the 200 J of KE increase. We just assume these quantities. Once we have these quantities (in this case, simply by assumption; in other problems we will often find these quantities from other information), they dictate the PE increase. There are a number of ways to think about this intuitively. For example: If I do 300 J of work on an object, then if my force is the only force acting on it, the its KE will increase by 300 J. If I do more than 300 J of work, but friction reduces the net force on the object to 300 J, then KE will increase by 300 J. If all the nonconservative forces together (e.g., my force plus frictional force) do 300 J of work on the system, and if no other forces act, then the KE will increase by 300 J. If all nonconservative forces together do 300 J of work and the KE increases by only 200 J, then the nonconservative forces can't be the whole story, because the work done by the net force is equal to the change in KE. The conclusion is that other forces must also be acting, and since they aren't nonconservative (we've assumed that all nonconservative forces together are accounted for in that 300 J), those forces must be conservative. And they must do -100 J of work on the system, so that the net force does 300 J - 100 J = 200 J of work. Of course we can also resort to equations. Since `dW_NC_on = `dPE + `dKE, it follows that `dPE = `dW_NC_on - `dKE = 300 J - 200 J = 100 J. Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qClass notes #10. Why does it make sense that the work done by gravity on a set of identical hanging washers should be proportional to the product of the number of washers and the distance through which they fall? Why is this consistent with the idea that the work done on a given cart on an incline is proportional to the vertical distance through which the cart is raised? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Work is equal to the force times the distance travelled. The force on the system is the number of washers hung from the car times the force of gravity; this is how much gravity is pulling on the car. Then you multiply this by the distance moved by the washers. Both sides of the equation are equal and proportional. When a car is pulled up an incline, the distance the washers move is equal to the distance the car moves vertically. This means that the equation will be the same for this example: the vertical distance times the force of the washers pulling down. Confidence rating #$&*: 2
.............................................
Given Solution: `a** Informally: The more clips, the more gravitational force, and the more the clips descend the more work is done by that force. The amount of work depends on how many clips, and on how far they descend. The number of clips required is proportional to the slope (as long as the slope is small). More formally, the force exerted by gravity is the same on each clip, so the total gravitational force on the hanging clips is proportional to the number of clips. The work done is the product of the force and the displacement in the direction of the force, so the work done is proportional to product of the number of washers and the vertical displacement. To pull the cart up a slope at constant velocity the number of washers required is proportional to the slope (for small slopes), and the vertical distance through which the cart is raised by a given distance of descent is proportional to the slope, to the work done is proportional to the vertical distance thru which the cart is raised. ** Self-critique (if necessary): The number of washers is proportional to the slope and the vertical distance is proportional to the slope, so the number of washer is proportional to the vertical distance. Self-critique rating #$&* 3 ********************************************* Question: `qHow does the work done against friction of the cart-incline-pulley-washer system compare with the work done by gravity on the washers and the work done to raise the cart? Which is greatest? What is the relationship among the three? Your solution: If you have a constant velocity system, the net force has to be 0, so the work done by gravity has to be equal to the other two forces together. The work done by gravity is the greatest. Also, the sum of the work against friction and the work against gravity will be equal to the work done by gravity on the car. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The force exerted by gravity on the hanging weights tends to move the system up the incline. The force exerted by gravity on the cart has a component perpendicular to the incline and a component down the incline, and the force exerted by friction is opposed to the motion of the system. In order for the cart to move with constant velocity up the incline the net force must be zero (constant velocity implies zero accel implies zero net force) so the force exerted by gravity in the positive direction must be equal and opposite to the sum of the other two forces. So the force exerted by gravity on the hanging weights is greater than either of the opposing forces. So the force exerted by friction is less than that exerted by gravity on the washers, and since these forces act through the same distance the work done against friction is less than the work done by gravity on the washers. The work done against gravity to raise the cart is also less than the work done by gravity on the washers. Work done against friction + work against gravity to raise cart = work by gravity on the hanging weights. ** Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qWhat is our evidence that the acceleration of the cart is proportional to the net force on the cart? Your solution: Force is equal to mass times acceleration. When you have more force on an object then the acceleration will be faster than on an object with equal mass and less force. Confidence rating #$&*: 3
.............................................
Given Solution: The net force is the sum of the gravitational force on the weights, and the frictional force (one force being positive, the other negative). The acceleration is the net force divided by total mass (mass of cart plus hanger plus washers). Washers are progressively transferred from the cart to the hanger, which keeps the mass of the system constant while increasing the net force. So the acceleration increases with the number of washers on the hanger. The gravitational force on the weights is therefore proportional to the number of washers on the hanger. With each added washer we get the same additional force, so we get the same additional acceleration. So the graph is linear. However the acceleration is not proportional to the number of weights. The net force on the system is equal to the gravitational force on the weights, plus the frictional force (which is of opposite sign, so while we are in fact adding quantities of opposite signs, it 'feels' like we're subtracting frictional force from gravitational force). The gravitational force on the weights is proportional to the number of washers, but when we add in the effect of the frictional force, our force is no longer proportional to the number of weights. Still linear, but not proportional to ... . Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qprin phy and gen phy prob 34: Car rolls off edge of cliff; how long to reach 85 km/hr? Your solution: 85km/hr(1000m/1km)(1hr/3600s) =23.61m/s 23.61m/s = 0km/h +9.8m/s^2 dt 9.8m/s^2 dt = 23.61m/s Dt = 2.41 seconds Confidence rating #$&*: 3
.............................................
Given Solution: `aWe know that the acceleration of gravity is 9.8 m/s^2, and this is the rate at which the velocity of the car changes. The units of 85 km/hr are not compatible with the units m/s^2, so we convert this velocity to m/s, obtaining velocity 85 km/hr ( 1000 m/km) ( 1 hr / 3600 sec) = 23.6 m/s. Common sense tells us that with velocity changing at 9.8 m/s every second, it will take between 2 and 3 seconds to reach 23.6 m/s. More precisely, the car's initial vertical velocity is zero, so using the downward direction as positive, its change in velocity is `dv = 23.6 m/s. Its acceleration is a = `dv / `dt, so `dt = `dv / a = 23.6 m/s / (9.8 m/s^2) = 2.4 sec, approx.. Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `q**** prin phy and gen phy problem 2.52 car 0-50 m/s in 50 s by graph How far did the car travel while in 4 th gear and how did you get the result? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: V0 = 36.5m/s vf = 45m/s Dt = 11.5s vAve = 81.5 / 2 = 40.75m/s ds = 40.75m/s*11.5s = 468.625m confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** In 4th gear the car's velocity goes from about 36.5 m/s to 45 m/s, between clock times 16 s and 27.5 s. Its average velocity on that interval will therefore be vAve = (36.5 m/s + 45 m/s) / 2 = 40.75 m/s and the time interval is 'dt = (27.5s - 16s) = 11.5 s. We therefore have 'ds = vAve * `dt = 40.75 m/s * 11.5 s = 468.63 m. The area under the curve is the displacement of the car, since vAve is represented by the average height of the graph and `dt by its width. It follows that the area is vAve*'dt, which is the displacement `ds. The slope of the graph is the acceleration of the car. This is because slope is rise/run, in this case that is 'dv/'dt, which is the ave rate of change of velocity or acceleration. We already know `dt, and we have `dv = 45 m/s - 36.5 m/s = 8.5 m/s. The acceleration is therefore a = `dv / `dt = (8.5 m/s) / (11.5 s) = .77 m/s^2, approx. ** Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `q **** Gen phy what is the meaning of the slope of the graph and why should it have this meaning? Your solution: Slope is equal to the acceleration which is the change in velocity divided by the change in clock time. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The graph is of velocity vs. clock time, so the rise will be change in velocity and the run will be change in clock time. So the slope = rise/run represents change in vel / change in clock time, which is acceleration. ** Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qGen phy what is the meaning of the area under the curve, and why does it have this meaning? Your solution: The area is the displacement: it is the average of the legs (velocity) times the time interval. Confidence rating #$&*: 3
.............................................
Given Solution: `a** The area under the curve is the distance traveled. This is so because 'ds = vAve*'dt. 'dt is equal to the width of the section under the curve and vAve is equal to the average height of the curve. The area of a trapezoid is width times average height. Although this is not a trapezoid it's close enough that we for the purpose of estimation can analyze it as such. Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `qGen phy what is the area of a rectangle on the graph and what does it represent? Your solution: The rectangle represents distance travelled. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** The area of a rectangle on the graph represents a distance. ** Self-critique (if necessary): OK Self-critique rating #$&* OK ********************************************* Question: `quniv phy problem 2.90 from 10th edition (University Physics students should solve this problem now). Superman stands on the top of a skyscraper 180 m high. A student with a stopwatch, determined to test the acceleration of gravity for himself, steps off the top of the building but Superman can't start after him for 5 seconds. If Superman then propels himself downward with some init vel v0 and after that falls freely, what is the minimum value of v0 so that he catches the student before that person strikes the ground? `quniv phy what is Superman's initial velocity, and what does the graph look like (be specific)? Your Solution: Student: 180m = 0m/s dt +.5 (9.8m/s^2)(dt^2) 108m = 4.9m/s^2 dt^2 Dt = 6.06s until he hits the ground Superman: 5 seconds less to get there, so he has 1.06s -180m = v0(1.06) +.5(9.8m/s^2)(1.06)^2 -180m = 1.06 v0 +5.50564m 1.06 v0 = -185.506m V0 = -175m/s Confidence rating #$&* 3
.............................................
Given Solution: `a** In time interval `dt after leaving the building the falling student has fallen through displacement `ds = v0 `dt + .5 a `dt^2, where v0 = 0 and, choosing the downward direction to be positive, we have a = -9.8 m/s^2. If `ds = -180 m then we have `ds = .5 a `dt^2 and `dt = sqrt(2 * `ds / a) = sqrt(2 * -180 m / (-9.8 m/s^2)) = 6 sec, approx.. Superman starts 5 seconds later, and has 1 second to reach the person. Superman must therefore accelerate at -9.8 m/s^2 thru `ds = -180 m in 1 second, starting at velocity v0. Given `ds, `dt and a we find v0 by solving `ds = v0 `dt + .5 a `dt^2 for v0, obtaining v0 = (`ds - .5 a `dt^2) / `dt = (-180 m - .5 * -9.8 m/s^2 * (1 sec)^2 ) / (1 sec) = -175 m/s, approx. Note that Superman's velocity has only about 1 second to change, so changes by only about -9.8 m/s^2, or about -10 m/s^2. ** ``qsketch a graph of Superman's position vs. clock time, and on the same graph show the student's position vs. clock time, with clock time starting when the person begins falling Your solution: Position is on the y axis and time is on the x. The students graph begins on the y axis at 180m. The curve decreases at an increasing rate because the velocity is increasing due to acceleration. The curve hits the x axis at x=6.06seconds. Supermans graph begins in the middle of the graph, at x=5. The slope is very steep downward. The curve intersects the students line right at its x-intercept. Both of the curves are parabolas and they are both concave down because the slope is increasing. Confidence rating #$&*: 3
.............................................
Given Solution: `a** If we start our clock at t = 0 at the instant the student leaves the top of the building then at clock time t the student's `dt will be just equal to t and his position will be x = x0 + v0 t + .5 a t^2 = .5 a t^2, with x0 = 180 m and a = -9.8 m/s^2. A graph of x vs. t will be a parabola with vertex at (0,180), intercepting the t axis at about t = 6 sec. For Superman the time of fall will be `dt = t - 5 sec and his position will be x = x0 + v0 (t-5sec) + .5 a (t-5sec)^2, another parabola with an unspecified vertex. A graph of altitude vs. t shows the student's position as a parabola with vertex (0, 180), concave downward to intercept the t axis at (6,0). Superman's graph starts at (5,180) and forms a nearly straight line, intercepting the t axis also at (6,0). Superman's graph is in fact slightly concave downward, starting with slope -175 and ending with slope -185, approx. ** Self-critique (if necessary): OK Self-critique rating #$&* OK "