________________: Find the acceleration of a block of mass m on a ______________ ramp that makes an angle q with the horizontal. Inclined Plane frictionless FN a a = Fnet/m w = mg q Fnet To determine a, you need ______, which means you need the ___________ that act on the block. forces

To find the force that accelerates the block down the incline, you need to choose new axes that are ______________ ( ) and __________________( ) to the plane’s surface. parallel || perpendicular Notice: The incline angle ____ is _________ as this one. New Axes: perpendicular q the same FN parallel || q || w q

The__________________ and _________________ components of ____ are the sides of a ___________ triangle: perpendicular parallel right w w|| adjacent side q || w┴ hypotenuse opposite w side w┴ adj hyp = = ______ cosq  w┴ = ________ wcosq w w|| opp hyp w|| = _________ wsinq = = _______ sinq  w

Now return to the full free body diagram and replace ____ by its ________________. FN w w|| = ________ components wsinq w┴ = _______ q || wcosq w FN FN ____ is balanced by _____:  ___ = _________ w┴ wcosq ___ is unbalanced and equals Fnet = ______________ w|| wsinq

So what is the acceleration a? w sinq mg sinq a = Fnet m _______ m _______ m = = = gsinq mass Notice that the result is independent of ________ . Ex: Find the acceleration of a 0.4-N block down a frictionless incline if the angle is 100. Test: release from rest: d = vit + (1/2)at2 d = a = a = gsinq a = (9.81m/s2)sin100 (1/2)at2 2d/t2 a = 1.7 m/s2

To see if the equation a = _________ makes gsinq To see if the equation a = _________ makes sense for the incline, see what happens when q has the "_______________ " values: extreme Case 1: What is a if q = 900? a = gsinq _____ free fall gsin900 = 900 g = g ·1 = Case 2: What is a if q = 00? a = gsinq = gsin00 = constant v

Now solve the problem, but with the block _________ or moving with __________________due to _____________. at rest constant v friction, Ff. FN FN Ff Ff w|| q w┴ w w q If the block is not moving, Fnet = ___ , which means all forces must ____________________ . be balanced FN = ____ = _________ (equilibrium as before). w┴ wcosq Ff = ____ = _________ (now in __________________). w|| wsinq equilibrium

Ex. A 3.0-kg block rests on an incline that makes an angle 300 to the horizontal. Determine the magnitude of the normal force and the friction force acting on the block. 3(9.81) w = ____ = __________ __________ mg 29 N 300 w┴ wcosq FN = ____ = _________ = _____________ = __________ (29 N)cos300 25 N Ff = ____ = _________ = _____________ = __________ w|| wsinq (29 N)sin300 14.5 N

C. Summary of inclined plane basics Without friction: With friction and at rest or constant v: FN FN Ff w|| w|| q q w┴ w┴ w w w┴ wcosq w┴ wcosq same FN = ____ = _________ FN = ____ = _________ Ff = ____ = _________ w|| w|| wsinq Fnet = ____ = _________ wsinq ___ = _________ a gsinq Fnet = __

Ex: If there is friction, but the block is ____________________ , it must be that _____ > _____ . In that case, the net force Fnet = ________ FN accelerating w|| Ff Ff w|| q w|| - Ff w┴ w Ex: Notice that the incline __________ or weakens gravity because it is not _______________ , but only the ________________________ that is pulling the block along the __________ of the incline. dilutes w itself parallel component of w w|| surface w

Final notes: 1/ On a horizontal surface: FN = ____ w BUT, On an incline (either with or w/o friction): FN FN = w┴ = ______ and NOT: FN = ____ wcosq w|| q w┴ w w

2/ What happens to Ff and FN if q is increased and the block remains the same and at rest? Ff FN w┴ w|| FN original incline: w Ff w|| q w┴ w inc. q q decreases FN __________ Ff ___________ w ____________ As _____ increases: q increases is the same