P. O. D. Station 2. You already have the real time. You found that with your stop watch.

P. O. D. Station 2 In Station 2 you have to find the real time (t real ), the real acceleration (a real )and the real force (Force real ). Then you have to find the ideal force, the ideal acceleration, and the ideal time. You already have the real time. You found that with your stop watch.

P. O. D. Station 2 How do we find the real acceleration of the car? Well, we know that according to Newton s 2 nd Law: F = ma. To find the force we need the mass and acceleration of the car. We know the mass, because we measured it (I will assume the mass of the car was 0.25 kg) To find the acceleration we need to use a formula that we saw first semester: distance = initial velocity time + ½ acceleration time 2 I will assume the ramp is 0.4 m long (distance), and it took the car an average of 0.3 sec to go down the ramp (time) x = v xo t + ½at 2 x = 0 +½at 2 Since there is no initial velocity 0.40 m = ½a(0.30 s) 2 Plugging in our known values 0.40 m = ½a(0.09) 0.40 m = 0.045a 0.045 0.045 8.89 m/s 2 = a real

P. O. D. Station 2 How do we find the real Force (F real ) that makes the car slide down the inclined plane? Now that we have the acceleration, let s find the REAL force using the formula: F = ma. F real = (0.25 kg)(8.89 m/s 2 ) F real = 2.22 N Now we have found the real time it takes to go down the slide, the real acceleration of the car, and the real force pushing it down the slide. But in the real world, friction exists. How would things have turned out w/ no friction, in other words, in the ideal world? Would t real < t ideal or would t real > t ideal? Would a real < a ideal or would a real > a ideal? Would F real < F ideal or would F real > F ideal?

What force really causes the car to go down the inclined plane if there is no hand or wind behind the car pushing it down the slide? ANS. It is the force due to gravity. The weight of the car makes it slide down. But wait a second. F w points downward ( ), but the car slides diagonally ( )down the inclined plane... F w

F kick F y F x Part of the kick (F kick ) makes the bag move left (F x ) but part also makes the bag want to move up (F y ).

F x is the horizontal component of the Force F w. F y is the vertical component of the Force F w. F x is the force that moves the car down the inclined plane. F y is the force that presses the car into the inclined plane. How do I find the ideal force that moves the car down the inclined plane? F y F x F y F w F x F w

TRIGONOMETRY 101 Fy Fx = opp. Fw = hyp. There are 2 right triangles in the Figure at left. If we look carefully, we can see that these 2 triangles are similar triangles. That means that the relationships between the sides and angles are the same in the two triangles. By the laws of trigonometry, the hypotenuse of the triangle, (Fw) is related to the side of the right triangle (Fx) that moves the car down the inclined plane by this formula: sin = opp. /hyp. sin = Fx /Fw Fwsin = Fx

opposite F x = F w sin The formula for weight (F w ) is F w = m g. So F x = F w sin becomes F x = m g sin We know the values of m & g. How do we find angle? We have to measure our ramp: sin = adjacent Opp. Hypo. 0.38 m sin -1 sin -1 sin = 0.38 0.40 sin = 0.95 sin -1 (sin ) = sin -1 (0.95) = 71.81

Let s find the ideal force F x = F w sin The formula for the weight, F w, is mass x gravity. F x = F w sin F x = m g sin Plug in your values: F x = 0.25 kg 9.8 m/s sin (71.81 ) F x = 2.33 N In the ideal world, F x, is the force that moves the cart down the plane, it is F ideal. So, F x = F ideal = 2.33 N Notice that F ideal (2.33 N) > F real (2.22 N) as predicted

Now, find the ideal acceleration using the force (F ideal ) and acceleration (a ideal ) of the car. F = m a F ideal = m a ideal 2.33 N = 0.25 kg a ideal 2.33 = 0.25 a ideal 0.25 0.25 9.32 m/s 2 = a ideal Notice that a ideal (9.32 m/s 2 ) > a real (8.89 m/s 2 ) as predicted

To find the ideal time, again use the formula: x = ½at 2 & solve for t 0.40 m = ½ (9.32 m/s)t 2 ideal 0.40 m = 4.66t 2 ideal 0.40 m = 4.66t 2 ideal 4.66 4.66 0.08 = t 2 ideal 0.08 = t 2 ideal 0.29 sec = t ideal Notice that t ideal (0.29 sec) < t real (0.3 sec) as predicted

To find the force of friction, just subtract the Force real from the Force ideal F frict = F ideal F real F frict = 2.33 N 2.22 N F frict = 0.11 N

Station 3 F T F y F T is what you measured on your spring scale. F y is the same as F w, the weight of the mass that was hanging from the stand. F x F w

Station 3 F = 12 N T F y = 8 N F y = F w = m g = 0.8125 kg x 9.8 m/s 2 Once again, the relationship is sin = opp. /hyp. sin = F y /F T F T sin = F Y With my numbers: 12N sin = 8 N 12N sin = 8 N 12 N 12 N F w = m g sin = 0.66 sin 1 (0.6666) = sin 1 (sin ) 41.8º =

Station 5 F Y1 F Y2 F T1 F T2 F X1 F X2 F w

Station 5 F Y1 F Y2 F T1 F T2 F X1 F X1 F X2 F T1 F Y1 F w

F y F T1 F y1 F y2 F T2 F x1 F x2 F w

F y F y1 F y2 F T1 F T2 F x1 F x2 F w

F y Station 5 = 3.92 N F y = F w /2 = (m g)/2 = (0.8 kg x 9.8 m/s 2 )/2 Once again, the relationship is sin = opp. /hyp. sin = F y /F T F T sin = F Y With my numbers: 12N sin = 3.92 N 12N sin = 3.92 N F 12 N 12 N w sin = 0.33 F w = m g sin 1 (0.333) = sin 1 (sin ) 19.48 =

STATION 4: Inclin-o-meter F N The normal force going up is F N sponge The force of weight going down F w

STATION 4: Inclin-o-meter F f F y F N F N F x sponge The force of friction (F f ) grows to match F x and F N shrinks to match F y F w The force of weight going down

STATION 4: Inclin-o-meter F y F f F N F x sponge The Normal Force (F N ) just equals F y The force of friction (F f ) just equals the force down the plane (F x ) The force of weight going down

STATION 4: Inclin-o-meter F fs The Normal Force (F N ) just F Y F N sponge F N The force of friction becomes F x sliding friction (F fs ) and shrinks below F x and the sponge begins to slide down the plane. equals F y

STATION 4: Inclin-o-meter F N F Y F N, the normal force, equals F y, the vertical component of F W, the weight: F N = F y F f, the friction force, equals F x, the horizontal component of F W, the weight: F f = F x We already know that F x = F w sin F w We can assume that F y = F w cos So F f = F w sin, and F N = F w cos

STATION 4: Inclin-o-meter The formula for the coefficent of friction F Y = F f F w F N F X = is Force of Friction = Normal Force And if we substitute from the previous slide, lo and behold F w sin F w sin = = = F N F Y F w cos F w cos = tan The coefficient of friction,, is simply tan. F f F N