Steven W. answered • 10/14/16

Tutor

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(2,279)
Physics Ph.D., college instructor (calc- and algebra-based)

Hi Jacob!

a) I will assume that means the ramp is inclined at 25° above the horizontal. In this case, if we place the 50 N sofa on the incline, it turns out to feel a force down the incline of (50 N)sin(25

^{o}), due to the component of gravity that is parallel to the incline's surface.As it attempts to slide down the plane under the influence of this force, static friction tries to hold it back. The maximum force static friction can muster is: F

_{s-max}= μ_{s}N, where N is the normal force of the plane on the sofa.We need to know the value of N to determine the magnitude of F

_{s-max}. To determine N, we look in the direction in which it acts: perpendicular to the plane. In the perpendicular direction, there is the component of the force of gravity perpendicular to the plane ((50 N)cos(25^{o})) and the normal force. So:Fnet-perp = N - (50 N)cos(25

^{o}) = 0 (since the sofa is stationary along a line perpendicular to the plane)N = (50 N)cos(25

^{o})This means F

_{s-max}= (0.6)(50 N)cos(25^{o}), and this is the maximum force of friction that can act back UP the plane, to prevent the sofa from sliding.So now, for the net force parallel to the plane (with down the plane being negative), we have:

F

_{net}= -(50 N)sin(25^{o})+(0.6)(50 N)cos(25^{o}) =and we need to add one more force (let's call it F

_{p}) to assure that the net force is zero, so that the sofa does not slide down the plane. Thus, we need:-(50 N)sin(25

^{o}) + (0.6)(50 N)cos(25^{o}) + F_{p}= 0 (we can safely assume F_{p}is up the plane, since we need to hold the sofa up)This is an equation you can solve for F

_{p}, the magnitude of the force you have to add to hold up the sofa, which is what this part is asking for.*********************************************************************************************

b) If the sofa is moving up the incline with constant speed, there are two changes to the previous net force equation:

-- that the friction force is now kinetic, rather than static, friction, since the sofa is now sliding across the plane surface

-- and that friction now points down, rather than up, the plane, because the sofa is now not trying to slide down the plane; it is being moved up the plane, and friction always points opposite to motion (or attempts at motion)

So the net force expression parallel to the plane now becomes (including this new pulling force up the plane, F

_{p}, we are providing):F

_{net}= F_{p}- μ_{k}N - (50 N)sin(25^{o}) = ma = 0 (because the sofa is moving at constant velocity, which means acceleration is 0)F

_{p}- (0.6)(50 N)cos(25^{o}) - (50 N)sin(25^{o}) = 0This is another equation that can now be solved for F

_{P}. This value for F_{p}will be the answer for Part b)*********************************************************************************************

Both of these answers for a) and b) assume F

_{p}is parallel to the plane, as stated in the problemFor c), just plug F

_{p}= 94 N into the force equation from Part b), and solve for the acceleration that would result (you are no longer assuming a = 0, as in part b); so, instead of knowing a=0 and solving for the F_{p}that would make that true, you are knowing F_{p}and solving for the acceleration that would result)Part d) is a kinematics expression in the direction parallel to the plane, but I think we may need one more piece of information that we have. Because we have the situation:

to find: (x-x

_{o}) (displacement)know: a (from Part c), t (given),

but we need to know three other kinematic quantities to solve for a fourth, and our choices are either initial velocity or final velocity. We have to know one of those. Does it say the couch is pulled with this acceleration "from rest," perhaps? That would mean initial velocity = 0. But without knowing either initial or final velocity of the sofa, I am not sure how to complete the problem.

If we assume the couch starts at rest and is pulled with that constant acceleration the whole time, then we can use:

(x-x

_{o}) = v_{o}t + (1/2)at^{2}to solve for the displacement.

I hope this helps get you started! If you have any more questions or want to talk further, just let me know.

Steven W.

tutor

I wondered the same thing, Arturo, though I suppose it is not beyond the pale for then to be equal. But the other thing that struck me was the 50 N weight for the sofa. That means its mass is only about 5 kg. That is one light sofa!

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10/14/16

Arturo O.

Yes, it is a light sofa! Maybe the problem should have stated a "block" instead of a sofa.

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10/14/16

Arturo O.

10/14/16