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PHY 201
Your 'cq_1_17.2' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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SEED Question 17.2
A 5 kg cart rests on an incline which makes an angle of 30 degrees with the horizontal.
• Sketch this situation with the incline rising as you move to the right and the cart on the incline. Include an x-y coordinate system with the origin centered on the cart, with the x axis directed up and to the right in the direction parallel to the incline.
The gravitational force on the cart acts vertically downward, and therefore has nonzero components parallel and perpendicular to the incline.
Sketch the x and y components of the force, as estimate the magnitude of each component.
What angle does the gravitational force make with the positive x axis, as measured counterclockwise from the positive x axis? Which is greater in magnitude, the x or the y component of the gravitational force?
answer/question/discussion: ->->->->->->->->->->->-> :
Given: m = 5 kg
Angle (theta) = 30 degrees
180 degrees + 40 degrees = 240 degrees
The y components will have the greater magnitude
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• Using the definitions of the sine and cosine, find the components of the cart's weight parallel and perpendicular to the incline.
answer/question/discussion: ->->->->->->->->->->->-> :
Fg (x) = Fg * sin 30 deg = mg * sin 30 deg = 5 kg * 9.8 m/s^2 * ½ = 24.5
Fg (y) = Fg * cos 30 deg = mg * cos 30 deg = 5 kg * 9.8 m/s^2 * 0.86 = 42.14
@& Your answers have the right magnitude, but the directions of your components are not specified.
The angle with the x axis is 240 degrees.
When using the angle with the positive x axis, you always have
F_x = F cos(theta)
F_y = F sin(theta).
This convention specifies the coordinate system and always returns the correct sign on the components relative to that system. We get
F_x = -24.5 N
F_y = -42.1 N.
Your solution was based on formulas that do not consistently the signs right. Those formulas are based on analysis by right triangles, and require that you explicitly specify your positive x and y directions, and then report the correct signs.
Each approach has its advantages and disadvantages. You are welcome to use either, but you need to understand the former apporoach, which is based on the more general circular definition of the sine and cosine functions. . I use that approach pretty much exclusively, for the following reasons:
1. Once the coordinate system is set up the signs take care of themselves.
2. Students with a marginal background in trigonometry can learn this system more easily than the triangle-based approach.
3. The circular definition is necessary for later work with gravitation, rotational motion, simple harmonic motion, wave motion and electrical circuits.
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• How much elastic or compressive force must the incline exert to support the cart, and what is the direction of this force?
answer/question/discussion: ->->->->->->->->->->->-> :
F(n) = Fg(y) = equal force (elastic and compressive force) will be exerted to support the cart. The direction of the force is downward.
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• If no other force is exerted parallel to the incline, what will be the cart's acceleration?
answer/question/discussion: ->->->->->->->->->->->-> :
F = m * a
a = F / m = Fg(x) / m= m g sin 30 deg / m = 9.8 m/s^2 * ½ = 4.9 m/s^2
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30 mins
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SEED Question 17.2 submitted 6 Mar 11 around 2:42 PM.
@& Good, but you didn't specify your coordinate systems or the directions of your components.
See my notes and also check the link below. No revision necessary unless you have questions.
&#See any notes I might have inserted into your document, and before looking at the link below see if you can modify your solutions. If there are no notes, this does not mean that your solution is completely correct.
Then please compare your old and new solutions with the expanded discussion at the link
Solution
Self-critique your solutions, if this is necessary, according to the usual criteria. Insert any revisions, questions, etc. into a copy of this posted document. Mark any insertions with &&&& so they can be easily identified.If your solution is completely consistent with the given solution, you need do nothing further with this problem. &#