Forces on an inclined plane Section 2.2
Examples of inclined planes
Previous Knowledge Since some of the incline problems lets review friction (both kinetic and static) and normal force.
Friction Friction is the force that hinders movement. f Ffriction It acts in the opposite direction to the applied force. The frictional force is dependent on the force keeping the objects together (F N ) and on the materials involved. 7
(F N ) Normal Force Normal force is the force that keeps surfaces together. It acts perpendicular to the surface. If the surfaces are horizontal, the normal force is equal to the force of gravity acting on the object. F N F g 8
On an incline F g F N <F g 9
Determine the normal force in each of the following cases. In each case the box has a mass of 10.0 kg A) B) 30 o 55 N 10
C) 75 N 70 o D) F = 1.2 N 11
E) F = 102 N 12
Formula for friction k f F N where is the coefficient of kinetic friction. It is a measure of the stickiness of the materials involved. f F N k is the frictional force. is the normal force. 13
Examples: 1. A box that has a mass of 48 Kg, is pulled along a horizontal surface by a rope that makes an angle of 30.0 with the horizontal. The tension in the rope is 90.0 N. If the coefficient of Kinetic friction is 0.025, A) what is the force of friction?
B) find the acceleration of the box?
2. Block A has a mass of 3.0 kg and Block B of 2.0 kg. They are connected by a string of negligible mass over a frictionless pulley. A A) If there is no friction, find the acceleration of B the system.
B) If the coefficient of kinetic friction is 0.10, find the acceleration of the system.
C)If the acceleration of the system is 2.5 m/s 2 (i) what is the frictional force between Block A and the table?
(ii) what is the coefficient of kinetic friction?
Static Friction Friction that prevents things from moving. It acts parallel to the 2 surfaces involved. To make an object slide you must first overcome static friction. Once the object starts to slide, kinetic friction takes over. Kinetic friction is less than static friction. 20
Frictional Force (N) Object Starts to move Graph of Frictional Forces vs Applied Force Kinetic Friction Applied Force (N) 21
Formula for Static Friction s f F N where s is the coefficient of static friction. f is the frictional force. F N is the normal force. 22
Example: A 10.0 kg box rests on a horizontal floor. The coefficient of static friction is 0.40 The coefficient of kinetic friction is 0.30 Determine the force of friction acting on the box if the following horizontal external forces are applied: A) 0 N B) 10 N C) 20 N D) 38 N E) 40 N 23
How much force is required to get the box to move? In other words what is the maximum static friction? f F s s N f mg s s f 0.40 10. kg 9.81 N / kg s 39.2N A) 0 N B) 10 N C) 20 N D) 38 N E) 40 N 24
What is the acceleration of the box? Fnet net Forces F F f a F 40N 29N 11N net a F net m 11 N m a 10 kg 1.1 s 2 25
The Ramp
Rotated coordinate system Motion on inclines Motion with friction on inclines
Inclined Planes Since many real life situations occur either on a ramp, hill, roof, or some other form of an incline, inclined plane problems are very common. Thus far we have looked at only horizontal (x) and/or vertical (y) motion.
Using this xy coordinate system for inclined plane situations can be problematic. This is because objects moving up (or down) an incline are moving in both the x and y direction at the same time. Therefore, a rotated coordinate system is used to solve incline plane problems. x-direction is rotated to be parallel to the plane s surface (called F in text) y-direction is rotated to be perpendicular to the plane s surface (called in text) F
Rotated Coordinate System y x Inclined Plane NOTE: Positive is taken to be in the direction of the motion
Show ALL of the Forces acting on the block below. F N f F g
What force makes the block slide down the incline? Gravity F N f F g
In fact it is the component of the force of gravity that acts parallel to the incline that makes the block slide down the ramp. This component is called F x or Is the normal force equal to the force of gravity? It is less than the force of gravity. The normal force is the component of the gravity (F y ) that acts perpendicular to the incline. F N = F y or F Thus F N and F are based on the mass of the block and the angle of the incline. F
What is the formula for F N and F? F N F = F N F =F N = Fgcosq =mgcosq F F F g F F = Fgsinq =mgsinq In this triangle where does q go? q WHY?
Examples Without friction: 1. Assuming a frictionless incline, find the acceleration of the 10. kg block below with incline angles: A) 0 o B) 30. o C) 45 o D) 60. o E) 90. o a F net F x m m mg sinq m g sinq
a Thus, acceleration is independent of mass in problems without friction Same as for free fall! A) 0.0 o B) 30.0 o C) 45.0 o a a 0 2 4.9 m 9.8 sin30 s m 2 s D) 60.0 o E) 90.0 o m 9.8 sin60 o 2 s 8.49 m s 2 o a a m 9.8 sin 45 o 2 s 6.93 m s 2 9.8 m 2 s
2. An object is travels up a frictionless incline (20.0 o to horizontal) with an initial speed of 4.00 m/s. Determine the times it takes for the object to come to a stop.
Examples with Friction Determine the acceleration on a block with mass m, on an incline with angle q, with a coefficient of sliding friction k. a F net F f m m f F mg a cosq N k k mg sinq mg cosq k gsinq gcosq k m
Thus, acceleration is independent of mass in problems with friction! EX. 3. Determine the acceleration on a block with mass 25.0kg, on an incline with angle 25.0 o, with a coefficient of sliding friction k 0.100 a g sinq g cosq k m o m o 9.8 sin25 9.8 cos25 (0.100) 2 2 s s 3.25 m s 2
In the previous example, find the time it would take for the block slide 30. 0cm if it starts sliding from rest.
Examples: 1. A book slides down a 4.0 m long ramp. If the ramp is inclined at an angle of 22 o to the horizontal and has a coefficient of kinetic friction of 0.12, how long will it take the 2.0 kg book to reach the bottom of the ramp?
2.A skateboarder slides down a frictionless ramp inclined at an angle of 30 to the horizontal. He then slides across a frictionless horizontal floor and begins to slide up a second incline at an angle of 25 to the horizontal. The skateboarder starts at a distance of 10 m from the bottom of the first incline. How far up the second incline will he go if the coefficient of kinetic friction on the second incline is 0.10?
3. Batman is driving the Batmobile down a hill coming from the Bat Cave. The hill is inclined at an angle of 30.0 to the horizontal and has a coefficient of kinetic friction of 0.28. What force must the Batmobile s engine apply to cause the Batmobile to accelerate at 0.60g?
Inclined planes can be used to determine coefficient of static friction The force of static friction is equal to the force of gravity acting along the plane, if the block is at rest. f F mg sinq And, s f s F N s x Also, F F mg cosq N mg sinq mg cosq y s tanq
The tan of the angle of the incline gives the coefficient of static friction. To find the maximum static friction, raise the incline until the object moves. Examples: 1. A ramp with a 45.0 kg box is raised until the box starts to slide. If this angle is of 42 o, what is the coefficient of static friction?
2. A 2.6m ramp with a 15.0 kg box placed at one end is raised until it starts to slide. The ramp's coefficient of static friction is 1.2 and the kinetic coefficient is 0.78. Determine the following: A) The speed of the box when it reaches the bottom of the ramp. B) The time it takes for the box to reach the bottom of the ramp.
,, 3. Given: mass of box = Find: a =? 5.0kg 0.17 k q 40.0 o q
,, 4. Given: mass of box= 0.50kg, a 2 1.70 m, 32.0 o s q. Find:?? k q
5. BF decides to hit the jump below. The coefficient of sliding friction on the in-run is 0.5 and 0.3 for the jump. The angle of the inrun is 32 o and 154m long. The angle of the jump is 22 o and 2.7m long. What is the horizontal distance from the end of jump to the spot where he lands into the water? The top of the jump is 7.4 m above the water and BF has a mass of 75 kg including snowboard gear.
Other Acceleration Problems Involving External Forces. Consider what happens when we either pull or push an object up, or down a ramp. Example: 1. Determine the acceleration if the 5.0 kg block is pulled up with a tension in the rope of 125 N k 0.40, q 25 o q
2. A)Determine the push required to accelerate a 45 kg child on a 2.6 kg toboggan to 0.70 g s (Down the Hill). k 0.42, q 32 o q
2. B)Determine the push required to accelerate a 45 kg child on a 2.6 kg toboggan to 0.70 g s (UP the Hill). k 0.42, q 32 o q
3. Two crates are dragged up a 10.0 o ramp with a force of 357 N. 0.52, m 35 kg, m 12kg k B s A)Determine acceleration of the boxes q
B) Determine the tension in the rope joining the two crates Solution: You need to consider either one of the two crates. (We ll do both in class) Draw a free body diagram for the object! THERE IS NO FORMULA TO REMEMBER! JUST FIGURE IT OUT
Crazy Table Problems 800. kg 4.00 kg k =0.0042 Find acceleration and tension a F net m a F f m m m 4.00kg 9.80 0.0042 800. kg 9.80 s s 804kg 0.00780 m s 2 a 2 2
Along comes Mr. Stickman If the incline of the table is 10 o, find the acceleration and tension a F net m F F f x g (4 kg box ) m k 4.00 kg =1.70m/s 2 m o m m 800kg 9.80 sin10 4.00kg 9.80 0.0042 800. kg 9.80 cos10 2 2 2 s s s 804kg o
Along comes Mr. Stickman If the incline of the table is 10 o, find the acceleration and tension 4.00 kg a=1.70m/s 2
Along comes Mr. Stickman, again a F net m F F f x g (4 kg box ) m k 4.00 kg If the incline of the table is 10 o, find the acceleration. Hint: Find the direction of motion first. =1.60m/s 2 (to the left) m o m m 800kg 9.80 sin10 4.00kg 9.80 0.0042 800. kg 9.80 cos10 2 2 2 s s s 804kg o
Along comes Mr. Stickman, again 4.00 kg Determine the tension. a=1.60m/s 2 (to the left)
Practice: 1. Two masses are connected together by a rope and pulley on a frictionless inclined plane as shown below. When the system is released, the 21 kg mass slides down the incline. What is the magnitude of the acceleration of the 21 kg mass? (A) 0.26 m/s 2 (B) 0.48 m/s 2 (C) 9.3 m/s 2 (D) 17 m/s 2 What is the tension in the rope?
2.Given the following diagram, and that the maximum tension that the rope can withstand is 150.0 N, determine: the tension in the string; and determine whether the string will break.
3. In the diagram below, what is the acceleration of m 2 down the incline? Include free body diagrams for m 1 and m 2. Friction is acting on the horizontal, but not on the incline. Find the tension in the rope.
4. Find the Tension =0.1 20 o 15kg
Double Incline Problems Level 3 question?? 60 o 50 o Find the acceleration without friction Hint: Find the direction of motion first.
Double Incline Problems 60 o 50 o Find the acceleration with friction = 0.01 (3 kg) = 0.02 (4 kg)
Double Incline Problems q 8 o Find the q such that there is no motion, and no friction.