course phy121
A 5 kg cart rests on an incline which makes an angle of 30 degrees with the horizontal. Sketch this situation with the cart on the incline and the x axis 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.
20N Parallel
29N perpendicular
Using the definitions of the sine and cosine, find are the components of the cart's weight parallel and perpendicular to the incline.
The weight parallel to the ramp is equal to the vertical force which is 5kg * 9.8m/s/s = 49N times the cos(30*).
49*cos(30*) = 42.44N
The perpendicular force is equal to the vertical force 49N times the sin(30*).
49*sin(30*) = 24.5N
The gravitational force does not add at an angle of 30 degrees with the positive x axis, so using cos(30 deg) for the x component and sin(30 deg) for the y component is not appropriate.
The coordinate system is rotated in such a way that the x axis is parallel to the incline. What therefore is the angle of the gravitational force, as measured counterclockwise from the positive x-axis?
This is the angle you need to use to calculate the components of the gravitational force.
How much elastic or compressive force must the incline exert to support the cart, and what is the direction of this force?
24.5N must be exerted to support the cart perpendicular to the ramp surface.
If no other force is exerted parallel to the incline, what will be the cart's acceleration?
Dividing the parallel force by the weight of the cart, we get:
42.44N / 5kg = 8.48m/s/s
You didn't use the correct angle; otherwise your reasoning was excellent. However do note that, even though your sketch underestimated the components, it did indicate that the parallel component was less than the perpendicular. Correcting the angle will take care of this inconsistency.
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