Asst 17 Query

course Phy 201

_ŕƬ{[~assignment #017

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017. `query 17

Physics I

03-16-2007

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16:32:19

ANSWERS/COMMENTARY FOR QUERY 17

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RESPONSE -->

ok

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16:39:02

prin phy and gen phy 6.33: jane at 5.3 m/s; how high can she swing up that vine?

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RESPONSE -->

`dKE = -`dPE

-.5 M v0^2 = -Mg * `dy

multiply both sides by -1 to get a positive

Divide both sides by Mg

`dy = (v0^2 / 2g) = (5.3 m/s / (2*9.8m/s/s))

`dy = 1.43 m

confidence assessment: 2

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16:39:14

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

-5 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.

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RESPONSE -->

ok

self critique assessment: 3

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16:46:54

prin phy and gen phy 6.39: 950 N/m spring compressed .150 m, released with .30 kg ball. Upward speed, max altitude of ball

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RESPONSE -->

PE elastic = .5kx^2 = .5 (950 N/m)(.0150 m) = 10.7 J

`dPE elastic = -10.7

`dPE grav = mgy = .30 kg * 9.8m/s/s * .150 m = .44 J

`dPE tot= -10.7 + 4.4 = -10.3 J

`dKE = -`dPE = 10.3 J

KEf = 10.3 J = .5 * .30 kg * v^2

Divide by .5* .30 kg

v = sqrt 68.7 = 8.3 m/s

PEgrav = mgy

y = PE / mg = 10.7 J / .30 KG * 9.8m/s/s

y = 3.64 m

confidence assessment: 2

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16:49:27

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.

The PE stored in the spring will be .5 k x^2 = .5 ( 950 N/m ) ( .150 m)^2 = 107 J. When released, conservation of energy (with only elastic and gravitational forces acting there are no nonconservative forces at work here) we have `dPE + `dKE = 0, so that `dKE = -`dPE.

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 t has a change in gravitational PE as well as elastic PE. The change in elastic PE is -107 J, and the change in gravitational PE is m g `dy = .30 kg * 9.8 m/s^2 * .150 m = +4.4 J. The net change in PE is therefore -107 J + 4.4 J = -103 J.

Thus between release and the equilibrium position of the spring, `dPE = -103 J

The KE change of the ball must therefore be `dKE = - `dPE = - (-103 J) = +103 J. The ball gains in the form of KE the 103 J of PE lost by the system.

The initial KE of the ball is 0, so its final KE is 103 J. We therefore have

.5 m vv^2 = KEf so that

vf=sqrt(2 KEf / m) = sqrt(2 * 103 J / .30 kg) = 26 m/s.

To find the max altitude to which the ball rises, we return to the state of the compressed spring, with its 107 J of elastic PE. Between release from rest and max altitude, which also occurs when the ball is at rest, there is no change in velocity and so no change in KE. No nonconservative forces act, so we have `dPE + `dKE = 0, with `dKE = 0. This means that `dPE = 0. There is no change in PE. The initial PE is 107 J and the final PE must also therefore be 107 J.

There is, however, a change in the form of the PE. It converts from elastic PE to gravitational PE. Therefore at maximum altitude the gravitational PE must be 107 J. Since PEgrav = m g y, and since the compressed position of the spring was taken to be the 0 point of gravitational PE, we therefore have

y = PEgrav / (m g) = 107 J / (.30 kg * 9.8 m/s^2) = 36.3 meters.

The ball will rise to an altitude of 36.3 meters above the compressed position of the spring.

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RESPONSE -->

why don't we write the distance the spring is compressed as -.150 since it is in the direction opposite to which the ball will be traveling?

It would be appropriate to write the displacement of the spring from equilibrium as -.150 m. However since the displacement is being squared, when calculating PE it isn't necessary to worry about this. The PE relative to equilibrium is always positive, except at equilibrium when it's 0.

Why don't we write gravity as -9.8m/s/s since it is in the direction opposite that in which the ball is traveling?

In a constant gravitational field `dPE = m g `dy, where m and g are scalar quantities rather than vector quantities (i.e., we use only the magnitudes of m and g). The one vector quantity is `dy, which is either positive or negative, depending on the situation.

The given solution is consistently off by a factor of 10--the 107 J should be 10.7 J, the 4.4 J should be .44 J and the change in altitude should be about 3.6 meters and not 36 meters.

Your solution was correct.

This was ridiculously complicated.

self critique assessment: 3

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18:50:32

gen 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?

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RESPONSE -->

`dPE = mgy = M* 9.8m/s/s * 2.1 m = M * 20.6 m^2/s^2

KEf = .5m v^2 = .5 * M * .49 m/s = M * .245 m^2/s^2

KE0 = .245 M m^2s^2 + 20.6 M m^2s^2 = 20.8 M m^2s^2

.5 V0^2 = 20.8 M m^2s^2

Divide both sides by .5

v0^2 = 41.6 m^2s^2

v = 6.4 m/s

confidence assessment: 2

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18:50:35

FORMAL 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.

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RESPONSE -->

self critique assessment: 3

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Good. See my notes and let me know if you have questions.