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:
240deg. Cos240= .33, sin240= .95, the y component is greater in magnitude
@& Good, but both functions are negative in the third quadrant, and the components would be obtained by multiplying these results by the weight (e.g., x component is -.33 * 49 N).*@
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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:
w parallel= cos240= .33, w perp= sin240= .95
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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:
.95N in the downward direction
@& The gravitational component perpendicular to the incline is in the negative y direction, so the incline will counter this with a force in the positive y direciton.*@
@& The y direction is more vertical than horizontal, but is not upward.*@
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• If no other force is exerted parallel to the incline, what will be the cart's acceleration?
answer/question/discussion:
.95N/5kg= .19m/s^2
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" "20min" ""
@& The x axis in in the direction parallel to the incline.
You have to multiply the sine and cosine by the 49 N weight to get the components of the gravitational force. The y component is balanced by the elastic or compressive force exerted by the incline; the x component is not countered by any force and therefore serves to acceleration the cart.
Check the notes at the link. I'm confident you'll understand everything, and if I'm right no revision is necessary.
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Then please compare your old and new solutions with the expanded discussion at the link
Solution
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