course Phy 241 瓙曋佼緪緥霈蝟帣鏊匋勩萤P醴撙assignment #002
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18:30:58 1) When masses of 35, 70 and 105 grams are hung from a certain rubber band its respective lengths are observed to be 39, 46, and 53 cm. What are the x and y components of the tension of a rubber band of length 50.6 cm if the x component of its length if 14.02232? What horizontal force, when added to this force will result in a total force of magnitude 150 grams?
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RESPONSE --> If the x component of the length is 14.02232 and the band length is 50.6 you can find the y component by using the pythogorean thereom. Therefore the y component of length is 48.6. The equation that relates the masses to the lengths is .2x+32. If we use this we can find the x component to correlate to a mass of 34.8 grams and a y component of 41.72 grams. The total force magnitude at this point can be represented 150^2=x^2 + 41.72^2, x is 144.08. To get to 144.08 from 34.8 we must add109.28 grams in the horizontal direction.
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18:34:07 STUDENT SOLUTION WITH INSTRUCTOR COMMENTS INDICATED BY ** I am assuming that you plug in what you know into the equation of the line. Y = .2x + 32 ** If y represents the mass and x the length that approach would give you the right mass--good thinking, though the equation would be more like y = 5 x - 150. Your equation looks reasonably good if y is length and x is supported mass. However you do need to use the force, which will be the force exerted by gravity on the mass. Force and mass are not the same thing. In general you will use the graph to determine the force corresponding to a given length. If there is significant curvature to the graph the linear model won?t work very well. In that case you could just draw a smooth curve to fit the data and read your forces from the curve. ** When the length is 50.6 cm the corresponding weight is 93 grams and when the length is 14.02232 cm the weight is ?89.9 g, but I don?t think weight can be negative???? ** Plug the rubber band length into the force relationship or read the force from your graph. If your graph is of supported mass vs. length you need to find the force. In this case you have a length of 50.6 cm, which corresponds to a supported mass of about 93 grams. This corresponds to a force of about .093 kg * 9.8 m/s^2 = .91 Newtons. The direction of the rubber band is arctan(y/x). Using the Pythagorean Theorem with length 50.6 cm and x component 14 cm we get y component about 48.6 cm, so the direction of the rubber band is arctan(48.6 / 14) = 74 deg. The angle at which the tension acts is parallel to the rubber band, so the tension also acts at 74 deg. The x and y components of the force are therefore Fx = .91 N * cos(74 deg) = .25 N and Fy = .91 N * sin(74 deg) = .87 N. If a horizontal force Fhoriz is added then the x component becomes Fx = .25 N + Fhoriz and the magnitude of the resulting force is | F | = sqrt(Fx^2 + Fy^2) = sqrt( (.25 N + Fhoriz)^2 + (.87 N)^2). If this force is 150 grams (or, converting this to a force, 1.47 N) then we have sqrt( (.25 N + Fhoriz)^2 + (.87 N)^2) = 1.47 N. Squaring both sides we get .0625 N^2 + .5 N * Fhoriz + Fhoriz^2 + .76 N^2 = 2.16 N^2. This is quadratic in Fhorz and rearranges to Fhoriz^2 + .5 N * Fhoriz ? 2.1 N^2 = 0. Using the quadratic formula we get Fhoriz = (-.5 N +- sqrt( (.5 N)^2 ? 4 * 1 * (-2.1 N^2) ) / (2 * 1) = (-.5 N +- 2.8 N) / 2, giving us two forces: Fhoriz = (-.5 N + 2.8 N ) / 2 = 1.15 N and Fhoriz = (-.5 N ? 2.8 N) / 2 = -1.65 N.**
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RESPONSE --> ok i needed to find the graph associated with lengths and forces using the masses given, this makes more sense.
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18:48:01 2) A ball of mass .3 kg is tossed vertically upward from altitude 4.1 meters and allowed to rise to a max altitude of 4.68 meters before falling to an altitude of 4.06 meters. * Determine the work done by gravity on the ball and the work done by the ball against gravity as it rises from initial to max altitude and determine its KE change during that displacement. * Using energy considerations determine its initial velocity. * Determine the work done by gravity on the ball and the work done by the ball against gravity from its initial all the way to the final position. ** Determine the change in KE between these positions and the use of energy considerations to determine its velocity at the final position. * How are the work done by the ball and its KE change related?
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RESPONSE --> The change in work is F`ds so we have F=ma or .3*-9.8 mulitplied by `ds of .58m we get `dW to be -1.71. This is the work done by gravity on the ball. The work done by the ball against gravity is 1.71. The KE change is also 1.71 if air resistance and thermal changes are neglected. At the maximum height the KE is 0 but the PE is 1.71, therefore the change was -1.71. If KE=mvf^2-mvi^2 and the initial velocity is 0 we can solve for vf as 2.39 m/s in the upward direction. From the maximum position to a change in `ds of .8m we can find the work done by `dW=ma`ds, if `ds is .8 and a is 9.8, and m is .3 then `dW is 2.35 N*m. If at the maximum height the `dW was 1.71 and PE was 1.71 then at the ending position `dW is -.64, this is associated with PE, therefore KE is .64. The velocity at the final position is therefore KE=mvf^2,vf is 1.46 in the downward direction.
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18:51:56 Intuitively the PE increases as the ball rises, so that KE decreases. Since we know that the KE of the rising ball is 0 at the top of the arc we can find the initial KE by finding the PE increase. As the ball falls, PE decreases and KE increases. Between initial and final position the ball has a small net downward displacement, hence a small net loss of PE, so it has a small net gain in KE. PE changes are easily found by multiplying the weight of the ball by the displacement. The detailed solution that follows is based on these concepts, but uses the work-energy theorem in a rigorous fashion: The work done by gravity on the ball is equal to the product of the force exerted by gravity, which is .3 kg * 9.8 m/s^2 = 2.94 N downward, and the displacement, which is 4.68 m ? 4.1 m = .58 m upward. This work is therefore -2.94 N * .58 m = -1.8 Joules, approx.. The work done by the ball against gravity is +1.8 J, equal and opposite to the work done by gravity on the ball. The work done against gravity is the change in the PE of the ball. Assuming no dissipative forces we have `dPE + `dKE = 0, so the KE change during this upward displacement is `dKE = -`dPE = -1.8 J. Since the final KE is 0, we have `dKE = KEf - KE0 = -1.8 J so that KE0 = 1.8 J + KEf = 1.8 J + 0 = 1.8 J. From initial to final position altitude changes by 4.06 m ? 4.1 m = -.04 m. Thus PE changes by -.04 m * 2.94 N = -.12 J. Since KE0 = 1.8 J and `dKE + `dPE = 0 we have `dKE = -`dPE = +.12 J so that KEf ? KE0 = .12 J and KEf = KE0 + 1.2 J = 1.8 J + .12 J = 1.92 J. The final velocity of the ball therefore satisfies .5 m vf^2 = KEf so that vf = +-sqrt(2 KE / m) = +- sqrt(1.92 J / (.3 kg) ) = +- sqrt( 6.4 m^2/s^2) = +- 2.5 m/s, approx.. Since final velocity is downward, our previous implicit choice of upward as the positive direction tells us that the final velocity is -2.5 m/s. The work done by the ball against gravity is equal and opposite to its change in KE.
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RESPONSE --> ok I found my change in work relative to the maximum height instead of comparing the starting position to the ending position of -.04. I am not sure why my calculations did not come out the same.
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18:53:58 3) Give an example of a situation in which you are given v0, a, and ?ds, and reason out all possible conclusions that could be drawn from these three quantities assuming uniform acceleration.
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RESPONSE --> If we use the equation vf^2=v0^2+2a`ds we can find vf and if we know vf we can find the `dv and vAve. If we know `dv and a, we can use a=`dv/`dt to find `dt.
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18:54:14 ** Your solution should show the use of the fourth equation of motion vf^2 = v0^2 + 2 a `ds to find vf. You then have the initial and final velocities, which since acceleration is uniform can be averaged to give you the average velocity vAve and the change `dv in velocity.. Dividing displacement `ds by average velocity we obtain `dt.
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RESPONSE --> ok
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18:59:32 4) A ball reaches a ramp of length 74 cm with an unknown initial velocity and accelerates uniformly along the ramp, reaching the other end in 5.2 seconds. Its velocity at the end of the ramp is 7.46154 cm/s. What is its acceleration on the ramp?
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RESPONSE --> We can use `ds=(vf+v0)/2*`dt to find a v0 of about 21.0cm/s. Now we can use `dv/`dt=a, -13.54 cm/s/5.2=-2.6cm/s^2
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18:59:49 STUDENT SOLUTION: I found the average velocity to be 74 cm / (5.2 s) = 14.23 cm/s. Since vAve = (vf + v0) / 2 we can easily solve for v0, obtaining v0 = 2 vAve ? vf = 2 * 14.23 cm/s ? 7.46 cm/s = 21 cm/s. Having vf and v0 we easily find that `dv = vf ? v0 = 7.46 cm/s ? 21 cm/s = -13.54 cm/s, and therefore a = `dv / `dt = -13.54 cm/s / (5.2 s) = ?2.6 cm/s^2
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RESPONSE --> ok
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19:10:32 5) A ball starting from rest rolls 10 cm down an incline on which its acceleration is constant, requiring .83 seconds to cover the distance. It then rolls onto a second incline 41 cm long on which its acceleration is 14 cm/s^2. How much time does it spend on the second incline and what is its acceleration on the first?
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RESPONSE --> First we use `ds=(vf+v0)/2`dt to find a vf of 24.1 cm/s. Now we can find an acceleration with `dv/`dt or 24.1/.83 is 29.04 cm/s^2. Now we can use `ds=v0`dt+.5a`dt^2. We get a quadratic, 7x^2+29x-41, x therefore is x is about 1.1 s.
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19:11:13 STUDENT SOLUTION: Ramp 1 I found the average velocity vAve = 10 cm / (.83 s) = 12 cm/s, approx.. Since v0 = 0 we have vf = 2 * vAve = 24 cm/s, approx. Acceleration on this incline is a = `dv / `dt = 24 cm/s / (.83 s) = 29 cm/s^2 approx. Ramp 2 I plugged in what I knew to find the unknowns, using v0 = 24 cm/s (the final velocity on the first incline is the init vel on the second), a = 14 cm/s^2 and `ds = 41 cm. I obtained vf = sqrt( v0^2 + 2 a `ds) = sqrt( (24 cm/s)^2 + 2 * 14 cm/s^2 * 41 cm) = 41 cm/s, approx.. Average velocity on the second incline is therefore (41 cm/s + 24 cm/s) / 2 = 32.5 cm/s, and the time required is `dt = `ds / vAve = 41 cm / (32.5 cm/s) = 1.3 sec, approx.
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RESPONSE --> ok
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19:31:28 6) A cart of mass 1.7 kg coasts 30 cm down an incline at 4 degrees with horizontal. Assuming Ffr is .042 times the normal force and other nongravitational forces parallel to the incline are negligible. * What is the component of the carts weight parallel to the incline? * How much work does this force do as the cart rolls down the incline? * How much work does the net force do as the cart rolls down the incline? * Using the definition of KE, determine the velocity of the cart after coasting the 30 cm assuming it initial velocity is zero.
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RESPONSE --> To find the parallel force you must take into accout `dh which is found by taking cos4 and multiplying it by .03cm. This comes to be .0299m. Multiplying that by m and g we get .499N. Work is found by taking .499 multiplied by .03 (`ds) or .015 N*m.
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19:41:10 The weight of the cart is 1.7 kg * 9.8 m/s^2 = 16.7 N, approx.. The incline makes an angle of 4 degrees with horizontal, so if the positive x axis is directed up the incline the weight will be at angle 270 deg ? 4 deg = 266 deg with the positive x axis and the x and y components of the weight will be x comp of weight = 16.7 N * cos(266 deg) = -1.2 N, approx., and y comp of weight = 16.7 N * sin(266 deg) = -16.6 N approx. The x component is parallel and the y component perpendicular to the incline. The normal force exerted by the incline is the elastic reaction to the y component of the weight and is +16.6 N. The parallel component of the gravitational force is in the direction of the displacement down the incline so the work done by this component on the cart is positive. We get work by parallel component of weight = 1.2 N * .30 m = .36 Joules, approx.. The frictional force is .042 times the normal force, or frictional force = .042 * 16.6 N = .7 N, approx., directed opposite to the displacement (up the incline). So the net force is net force = parallel component of gravitational force + frictional force = -1.2 N + .7 N = -.5 N, or .5 N down the incline. The work done by this force is .5 N * .3 m = .15 J. The KE of the cart after coasting down the incline from rest is by the work-energy theorem equal to the work done by the net force. So we have .5 m vf^2 = KE, so that vf = +- sqrt(2 KE / m) = +- sqrt( 2 * .15 J / (1.7 kg) ) = +- sqrt( .16 m^2/s^2) = +- .4 m/s, approx. Using the sign conventions specified by our choice of the upward direction as the positive x direction, the velocity will be -.4 m/s.
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RESPONSE --> I am not sure how we get the 266 degrees if the x axis is going up the incline.
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19:59:41 7) A ball mass 8 kg rolls off the edge of a ramp with a horizontal speed of 70 cm/s. * What is its KE as it rolls off the ramp? * How much work does gravity do on the ball as it falls 22 cm? * What will be its KE after falling 22 cm? *How much of this KE is accounted for by its horizontal velocity and how much by its vertical velocity? * What then is its vertical velocity at this point?
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RESPONSE --> KE=.5mv^2, therefore KE is 3.92J. We can find how much work gravity does on the ball as it falls by finding the Potential energy which is `dh(g)(a)or .22(9.8)(8)=17.25. The total KE will therefore be 17.25+3.92 or 21.17 J. Vertical velocity is found by taking KE=.5mv^2 or 21.17=.5(8)v^2, v=2.3m/s
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20:08:22 We have KE = .5 m v^2 = .5 * 8 kg * (.70 m/s)^2 = 2 Joules, approx.. As the ball falls 22 cm gravity does work `dWgrav = 8 kg * 9.8 m/s^2 * .22 m = 17 Joules, approx.. So after the 22 cm fall we will have KE = 2 J + 17 J = 19 J. There is no net force in the horizontal direction so the horizontal velocity will be unchanged and its horizontal KE will be 2 J, leaving 17 J of vertical KE. The vertical velocity will therefore be vertical velocity = +-sqrt(2 * vertical KE / m) = +- sqrt( 2 * 17 J / (8 kg)) = +-2.1 m/s.
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RESPONSE --> ok yes find the KE from the given velocity and mass, then we can find the Potential Energy in the vertical direction.
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