#$&* course PHY 201 4/12 about 8 pm 017. `query 17
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Given Solution: Outline of solution: Jane has KE. She goes higher by increasing her gravitational PE. Her KE is 1/2 m v_0^2, where m is her mass and v0 is her velocity (in this case, 6.3 m/s^2). If she can manage to convert all her KE to gravitational PE, her KE will decrease to 0 (a decrease of 1/2 m v0^2) and her gravitational PE will therefore increase by amount 1/2 m v_0^2. The increase in her gravitational PE is m g `dy, where m is again her mass and `dy is the increase in her altitude. Thus we have PE increase = KE loss In symbols this is written m g `dy = 1/2 m v0^2. The symbol m stands for Jane's mass, and we can also divide both sides by m to get g `dy = 1/2 v0^2. Since we know g = 9.8 m/s^2 and v0 = 6.3 m/s, we can easily find `dy. `dy = v0^2 / (2 g) which is easily evaluated to obtain `dy = 1.43 m. MORE DETAILED SOLUTION: Jane is going to convert her KE to gravitational PE. We assume that nonconservative forces are negligible, so that `dKE + `dPE = 0 and `dPE = -`dKE. Jane's KE is .5 M v^2, where M is her mass. Assuming she swings on the vine until she comes to rest at her maximum height, the change in her KE is therefore • `dKE = KEf - KE0 = 0 - .5 M v0^2 = - .5 M v0^2, where v0 is her initial velocity. Her change in gravitational PE is M g `dy, where `dy is the change in her vertical position. So we have `dKE = - `dPE, or - 1/2 M v0^2 = - ( M g `dy), which we solve for `dy (multiply both sides by -1, divide both sides by M g) to obtain `dy = v0^2 / (2 g) = (5.3 m/s)^2 / (2 * 9.8 m/s^2) = 1.43 m. STUDENT QUESTION: I’m confused as to where the 2 g came from INSTRUCTOR RESPONSE: You are referring to the 2 g in the last line. We have in the second-to-last line - 1/2 M v0^2 = - ( M g `dy). Dividing both sides by - M g, and reversing the right- and left-hand sides, we obtain `dy = - 1/2 M v0^2 / (M g) = 1/2 v0^2 / g = v0^2 / (2 g). STUDENT QUESTION do we get dy'=v0^2/2g will this always be the case? INSTRUCTOR RESPONSE Most basic idea: On the simplest level, this is a conversion of PE to KE. This is the first thing you should understand. The initial KE will change to PE, so the change in PE is equal to the initial KE. In this case the change in PE is m g `dy. For other situations and other conservative forces the expression for `dPE will be very different. The simplest equation for this problem is therefore init KE = increase in PE so that 1/2 m v0^2 = m g `dy More general way of thinking about this problem: More generally we want to think in terms of KE change and PE change. We avoid confusion by not worrying about whether each change is a loss or a gain. Whenever conservative forces are absent, or being regarded as negligible, we can set the expression for KE change, plus the expression for PE change, equal to zero. • In the present example, KE change is (final KE - initial KE) = (0 - 1/2 m v^2) = -1/2 m v^2, while PE change is m g `dy. • We get the equation -1/2 m v0^2 + m g `d y = 0. • This equation is easily rearranged to get our original equation 1/2 m v0^2 = m g `dy. The very last step in setting up the problem should be to write out the expressions for KE and PE changes. • The expression for PE change, for example, depends completely on the nature of the conservative force. For gravitational PE near the surface of the Earth, that expression is m g `dy. For gravitational PE where distance from the surface changes significantly the expression would be G M m / r1 - G M m / r2. For a spring it would be 1/2 k x2^2 = 1/2 k x1^2. • The expression for KE change is 1/2 m vf^2 - 1/2 m v0^2; this is always the expression as long as mass doesn't change. In this particular case the equation will read • 1/2 m vf^2 - 1/2 m v0^2 + m g `dy = 0 If we let vf = 0, the previous equations follow. STUDENT QUESTION Ok I understand gravitiaional PE=mgdy and I understand KE=.5mv^2 But Idont really understand how to combine the two equations. I can see that doing so allows us to eliminate m. since we don’t have that value. And I don’t see how KE and PE can cancel out to allow for the equations to combine. INSTRUCTOR RESPONSE In the absence of dissipative forces (e.g., friction) it is possible to convert KE to PE, or vice versa. Jane is moving at 5.3 m/s so she has KE. She wants to go higher so she converts her KE to PE by grabbing a vine. As she follows the resulting upward ard her PE increases at the expense of here KE. She continues rising until she has used up all her KE. At that point her KE will have been converted to PE. If her altitude increases by `dy, her PE will have increased by m g `dy. Her original KE was 1/2 m v0^2, where v0 is her original 5.3 m/s velocity. So we set m g `dy equal to 1/2 m v0^2, and solve for `dy. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: `qprin phy and gen phy 6.39: 950 N/m spring compressed .150 m, released with .30 kg ball. Upward speed, max altitude of ball YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: F=-.5*k*x ‘dw = ‘dPE_elastic = -.5*k*x*x = -.5*k*x^2 = -.5*950 N/m * (0.15 m)^2 = -10.7 J ‘dPE_g = m * g * ‘dy = .30 kg * 9.8 m/s^2 * 0.15 m = 0.44 J(this is positive because it is acting in the opposite direction of the compression of the spring, I think) ‘dPE_total = -10.7 + 0.44 J = -10.26 J = -10.3 J ‘dKE = -(‘dPE) = - (-10.3 J) = 10.3 J 10.3J = .5 m*v_f^2 - .5 m*v_0^2 = .5 m*v_f^2 V_f = ‘sqrt(10.3 J / (.5*m) = ‘sqrt(10.3 J / (.5*.30 kg)) = +/- 8.3 m/s = + 8.3 m/s This is actually the velocity at the end of the recoil of the spring which is equal to the initial velocity when all PE is converted to KE. V_f^2 = v_0^2 + 2*a*’ds ‘ds = (v_f^2 - v_0^2)/(2*a) = (0^2 - (8.3 m/s)^2)/(2 * -9.8 m/s^2) = 3.5 m confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a We being with a few preliminary observations: • We will assume here that the gravitational PE of the system is zero at the point where the spring is compressed. In this situation we must consider changes in both elastic and gravitational PE, and in KE. • We also observe that no frictional or other nonconservative forces are mentioned, so we assume that nonconservative forces do no work on the system. • It follows that `dPE + `dKE = 0, so the change in KE is equal and opposite to the change in PE. The PE stored in the spring will be .5 k x^2 = .5 ( 950 N/m ) ( .150 m)^2 = 10.7 J. Since the ball is moving in the vertical direction, between the release of the spring and the return of the spring to its equilibrium position, the ball has a change in gravitational PE as well as elastic PE. • The change in elastic PE is -10.7 J, and the change in gravitational PE is m g `dy = .30 kg * 9.8 m/s^2 * .150 m = +.44 J. • The total change in PE is therefore -10.7 J + 4.4 J = -10.3 J. Summarizing what we know so far: • Between release and the equilibrium position of the spring, `dPE = -10.3 J During this interval, the KE change of the ball must therefore be `dKE = - `dPE = - (-10.3 J) = +10.3 J. Intuitively, the ball gains in the form of KE the 10.3 J of PE lost by the system. The initial KE of the ball is 0, so its final KE during its interval of contact with the spring is 10.3 J. We therefore have • .5 m v^2 = KEf so that • vf=sqrt(2 KEf / m) = sqrt(2 * 10.3 J / .30 kg) = 8.4 m/s. To find the max altitude to which the ball rises, we consider the interval between release of the spring and maximum height. • At the beginning of this interval the ball is at rest so it has zero KE, and the spring has 10.7 J of elastic PE. • At the end of this interval, when the ball reaches its maximum height, the ball is again at rest so it again has zero KE. The spring also has zero PE, so all the PE change is due to the gravitational force encountered while the ball rises. • Thus on this interval we have `dPE + `dKE = 0, with `dKE = 0. This means that `dPE = 0. There is no change in PE. Since the spring loses its 10.7 J of elastic PE, the gravitational PE must increase by 10.7 J. • The change in gravitational PE is equal and opposite to the work done on the ball by gravity as the ball rises. The force of gravity on the ball is m g, and this force acts in the direction opposite the ball's motion. Gravity therefore does negative work on the ball, and its gravitational PE increases. If `dy is the ball's upward vertical displacement, then the PE change in m g `dy. • Setting m g `dy = `dPE we get `dy = `dPE / (m g) = 10.7 J / ( .30 kg * 9.8 m/s^2) = 10.7 J / (2.9 N) = 10.7 N * m / (2.9 N) = 3.7 meters. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):I did not initially take into account ‘dPE_g until I read your solution and then I reworked the problem. Also, I solved for ‘ds through uniform acceleration equations but I did understand how to solve it the way that you did. My answer for ‘ds was slightly different from your ‘dy but I think mine would still be correct. ------------------------------------------------ Self-critique rating:3
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Given Solution: Overall idea: The car will dissipate its kinetic energy as it runs up the hill, thereby increasing its gravitational potential energy as friction does negative work on it. It thus 'spends' its initial KE on the PE increase and the work it does against friction. Key formulation: `dW_noncons_ON = `dKE + `dPE, the work done ON the car by nonconservative forces is equal the net change in its mechanical energy (mechanical energy being the sum of its kinetic and potential energies). Important special case: If `dW_noncons_ON = 0 then `dKE + `dPE = 0, meaning that any change in kinetic energy is associated with an equal and opposite change in potential energy. Solution: (a) The car will coast up the hill until its kinetic energy has been partially dissipated against friction and partially converted to gravitational PE. If friction is negligible, then the KE will all be converted to gravitational PE. No mass is given for the car, so we will let m_car stand for its mass. The initial KE of the car is KE_0 = 1/2 m_car v_0^2 = 1/2 m_car (110 km / h)^2 = 1/2 m_car ( 110 (1000 m) / (3600 s) ) ^2 = 1/2 m_car * (30 m/s)^2 = m_car * 450 m^2 / s^2, where the 30 m/s is a rough approximation of 110 (1000 m) / (3600 s). We understand that gravitational PE increases with altitude. To take this back to the definition: The change in gravitational PE is equal and opposite to the work done by the gravitational force. Gravity exerts a downward force of magnitude m_car g, so if the car's vertical displacement is upward, gravity will do negative work and the PE change is therefore positive. If the upward direction is regarded as positive, then the force exerted by gravity (i.e., the weight of the car) is in the negative direction so will be represented by -m_car g. if the change in vertical position is `ds_y, then the work done by gravity is -(m_car g) * `ds_y and the change in gravitational PE is + m_car g `ds_y. The KE loss is therefore 1/2 m_car v_0^2 = m_car * 450 m^2 / s^2, which is equal to the increase in gravitational PE. So we can write m_car g `ds_y = m_car * 450 m^2 / s^2. Substituting 9.8 m/s^2 for g, we easily solve this equation for `ds_y and obtain `ds_y = 46 meters. It is preferable to also write and solve the same equation symbolically: m_car g `ds_y = 1/2 m_car v_0^2. Solving for v we first divide both sides by m_car to get g `ds_y = 1/2 v_0^2 so that `ds_y = 1/2 v_0^2 / g = 1/2 * (30 m/s)^2 / (9.8 m/s^2) = 46 (m^2 / s^2) / (m/s^2) = 46 (m^2 / s^2) ( s^2 / m) = 46 m. (b) If the car coasts to a point just 22 m higher than the original, then the change in its gravitational PE is `dPE = m_car * g * `ds_y = 750 kg * 9.8 m/s^2 * 22 m = 160 000 kg m^2 / s^2 = 160 000 Joules. Its initial KE was KE_0 = 1/2 m v_0^2 = 1/2 * 750 kg * (30 m/s)^2 = 340 000 Joules, approx. Its KE therefore changed by -340 000 Joules while its PE increased by 160 000 Joules. Intuitively, were no friction present the 340 000 Joules loss of KE would be accompanied by a 340 000 Joule increase in PE, but the PE change comes up 180 000 Joules short. This 180 000 Joules of energy was therefore dissipated by friction. In terms of the work-energy theorem: `dW_noncons_ON = `dKE + `dPE = -340 000 Joules + 160 000 Joules = -180 000 Joules. The nonconservative force at work here is friction ON the car, so `dW_noncons_ON = -180 000 Joules. That is, friction does -180 000 Joules of work on the car. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: `qgen phy problem A high jumper needs to be moving fast enough at the jump to lift her center of mass 2.1 m and cross the bar at a speed of .7 m/s. What minimum velocity does she require? Your solution: ‘dPE = -‘dKE M*g*’dy = -(.5*m*v_f^2 - .5 *m * v_0^2) = -.5*m*v_f^2 + .5*m*v_0^2 V_0^2 = (2*(m*g*’dy + .5 * m * v_f^2))/m = 2*g*’dy + v_f^2 V_0 = ‘sqrt(2*g*’dy + v_f^2) = ‘sqrt(2*9.8 m/s^2*2.1m + (0.7 m/s)^2) = 6.45 m/s = 6.5 m/s
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Given Solution: `aFORMAL SOLUTION: Formally we have `dPE + `dKE = 0. `dPE = M * g * `dy = M * 20.6 m^2 / sec^2, where M is the mass of the jumper and `dy is the 2.1 m change in altitude. `dKE = .5 M vf^2 - .5 M v0^2, where vf is the .7 m/s final velocity and v0 is the unknown initial velocity. So we have M g `dy + .5 M vf^2 - .5 M v0^2 = 0. Dividing through by M we have g `dy + .5 vf^2 - .5 v0^2 = 0. Solving for v0 we obtain v0 = sqrt( 2 g `dy + vf^2) = sqrt( 2* 9.8 m/s^2 * 2.1 m + (.7 m/s)^2 ) = sqrt( 41.2 m^2/s^2 + .49 m^2 / s^2) = sqrt( 41.7 m^2 / s^2) = 6.5 m/s, approx.. LESS FORMAL, MORE INTUITIVE, EQUIVALENT SOLUTION: The high jumper must have enough KE at the beginning to increase his PE through the 2.1 m height and to still have the KE of his .7 m/s speed. The PE change is M * g * 2.1 m = M * 20.6 m^2 / sec^2, where M is the mass of the jumper The KE at the top is .5 M v^2 = .5 M (.7 m/s)^2 = M * .245 m^2 / s^2, where M is the mass of the jumper. Since the 20.6 M m^2 / s^2 increase in PE must come at the expense of the initial KE, and since after the PE increase there is still M * .245 m^2 / s^2 in KE, the initial KE must have been 20.6 M m^2 / s^2 + .245 M m^s / s^2 =20.8 M m^s / s^2, approx. If initial KE is 20.8 M m^s / s^2, then .5 M v0^2 = 20.8 M m^s / s^2. We divide both sices of this equation by the jumper's mass M to get .5 v0^2 = 20.8 m^2 / s^2, so that v0^2 = 41.6 m^2 / s^2 and v0 = `sqrt(41.6 m^2 / s^2) = 6.5 m/s, appprox. STUDENT QUESTION I used the equation 'dy=v0^2 / (2g). Isn't that easier? INSTRUCTOR RESPONSE Good, but that equation only applies under certain conditions. Your solution didn't account for the final KE, which doesn't make a lot of difference but does make enough to decide the winner of a competitive match. In general you don't want to carry an equation like 'dy=v0^2/(2g) around with you. If you carry that one around, there are about a hundred others that apply to different situations, and you'll overload very quickly. Among other things, that equation doesn't account for both initial and final KE. It applies only when the PE change is gravitational, only near the surface of the Earth, and only when the final KE is zero. Way too many special conditions to keep in mind, way too much to remember. You want to start your reasoning from `dKE + `dPE + `dW_noncons_ON = 0. We assume that nonconservative forces are negligible, so that `dW_noncons_ON is itself zero, giving us `dPE + `dKE = 0. For this situation `dPE = m g `dy, `dKE = KE_f - KE_0 = 1/2 m vf^2 - 1/2 m v0^2, and the equation becomes m g `dy + 1/2 m v0^2 - 1/2 m vf^2 = 0. In a nutshell, there are only three things you need in order to analyze similar situations: • `dKE + `dPE + `dW_noncons = 0 • KE = 1/2 m v^2 • `dPE = m g `dy (in the vicinity of the Earth's surface) STUDENT QUESTION After looking throught the given solution I don’t see how any of the values were obtained. The only ones I understand is the dy=2.1 m and vf=.7 m/s Again I think a major problem for at this point is that I do not understand how to combine the equations. Ive read all of the notes and problem sets etc. However maybe I just didn’t take note of someof the things I should’ve at the time. If you tell me where I could look for that I think that would help me. Thanks. Your text should provide a good reference, as will the Introductory Problem Sets and qa's.. This problem is similar to the earlier problem. You can figure out the PE change as a multiple of the jumper's mass, using `dy = 2.1 m and g.= 9.8 m/s^2. You can figure out the jumper's KE as she goes over the bar at .7 m/s, again as a multiple of her mass. To get to the height of the bar and the .7 m/s velocity, she has to have KE equal to the sum of the PE gain and the KE at .7 m/s. So set 1/2 m v0^2 equal to the sum of the other two expressions and solve for v0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK Question (optional Openstax `prin, `gen): Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown. Your solution: -‘dPE = ‘dKE -(m*g*’dy) = .5*m*v_f^2 - .5*m*v_0^2 V_f^2 = (2*(-(m*g*’dy) + .5*m*v_0^2)/m V_f^2 = (-2*(g*’dy) + v_0^2) V_f = ‘sqrt((-2*(g*’dy) + v_0^2)) V_f=’sqrt(-2*(9.8 m/s^2*-20m) + (15 m/s)^2) = ‘sqrt(392 m^2/s^2 + 225 m^2/s^2) = ‘sqrt(617 m^2/s^2) = +/- 24.8395 m/s = + 24.8 m/s (positive since g was positive) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Main idea: The rock starts kinetic energy. Gravity does positive work on it as it falls, decreasing its potential energy and therefore increasing its velocity. Friction doesn't hinder the process. Formulation: `dW_noncons_ON = `dKE + `dPE. In this case the only nonconservative force would be friction, which however is assumed to be zero so that `dW_noncons_ON = 0. It follows that `dKE + `dPE = 0. Any change in KE or PE is accompanied by an equal and opposite change in the other. Solution: Gravity does positive work on the rock so its change in gravitational PE, being equal and opposite to the work done by gravity, is negative. We get `dPE = m_rock * g * `ds_y = m_rock * 9.8 m/s^2 * (-20 m) = -m_rock * 196 m^2 / s^2. It follows that `dKE = +m_rock * 196 m^2 / s^2, so that KE_f = KE_0 + `dKE = 1/2 m_rock * (24.8 m/s)^2 + m_rock * 196 m^2 / s^2 = m_rock * 310 m^2 / s^2 + m_rock * 196 m^2 / s^2 = m_rock * 500 m^2 / s^2, approximately. KE_f = 1/2 m_rock vf^2, so 1/2 m_rock vf^2 = m_rock * 500 m^2 / s^2 and vf = +- sqrt( 2 * 500 m^2 / s^2 ) = +-31 m/s, approx. A purely symbolic solution is more elegant: `dW_noncons_ON = 0 so, since `dW_noncons_ON = `dPE + `dKE we have `dKE + `dPE = 0 For this situation `dKE = 1/2 m_rock v_f^2 - 1/2 m_rock v0^2 `dPE = m_rock g `ds_y. Substituting into `dKE + `dPE = 0 we have 1/2 m_rock v_f^2 - 1/2 m_rock v0^2 + m_rock g `ds_y = 0 We divide both sides by m_rock to get 1/2 v_f^2 - 1/2 v0^2 + g `ds_y = 0 and solve for v_f to get vf = +- sqrt( 2 (1/2 v0^2 - g `ds_y)). Substituting for v0, g and `ds_y we have vf = +- sqrt( 2 * (1/2 * (24.8 m/s)^2 - 9.8 m/s^2 * (-20 m) ) = +- sqrt( 600 m^2 / s^2 + 400 m^2 / s^2) = +- sqrt( 1000 m^2 / s^2) = +- 31 m/s, approximately. We just want the speed of the rock, do we ignore the negative solution and conclude that the final speed is about vf = 31 m/s. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!