1. Forces in Two Dimension
Forces and Motion
Forces in Two Dimension

2. Vectors Vectors in Multiple Dimensions
We can add vectors even when they don’t point along same straight line
Draw the vectors at correct angle and measure direction and magnitude of resultant vector. Add vectors by placing them tip-to-tail and then drawing resultant of vector by connecting tail of first vector to tip of second vector

3. Vectors Vectors in Multiple Dimensions
If adding two vectors at right angles, use Pythagorean theorem to find magnitude of resultant, R
Sum of squares of magnitudes equal to square of magnitude of resultant vector

4. Vectors Law of cosines Law of sines Vectors in Multiple Dimensions
If two vectors at angle other than 90°, use law of cosines or law of sines
Square of magnitude of resultant vector equal to sum of magnitude of squares of two vectors, minus two times product of magnitudes of vectors, multiplied by cosine of angle between them
Magnitude of resultant, divided by sine of angle between two vectors, equal to magnitude of one vectors divided by angle between that component vector and resultant vector
Law of cosines
Law of sines

5. Vectors Vectors in Multiple Dimensions
Find magnitude of sum of 15-km displacement and 25-km displacement when angle between them 90° and when angle between them 135°

6. Vectors Components of Vectors
We start by choosing coordinate system similar to laying a grid on transparent sheet atop vector problem
Choose where to put center of grid (origin) and establish directions axes point

7. Vectors Components of Vectors
When motion on surface of Earth, x-axis east and y-axis north
When motion of object moving through air, positive x-axis horizontal and positive y-axis vertical (upward)
If motion on hill, place positive x-axis
in direction of motion and y-axis
perpendicular to x-axis
A = Ax + Ay A Ay Ax

8. Friction
Components of Vectors

9. Vectors Algebraic Addition of Vectors Ry = Ay + By + Cy
If you have two or more vectors, they are added by first resolving each vector into x- and y-components
X-components added to form x-component of resultant:
Rx = Ax + Bx + Cx
Y-components added to form y-component
of resultant:
Ry = Ay + By + Cy

10. Vectors Algebraic Addition of Vectors
The components of a vector are projections of the component vectors

11. Vectors Algebraic Addition of Vectors Angle of Resultant Vector
Rx and Ry at right angle (90°) so magnitude of resultant calculated using Pythagorean theorem, R2 = Rx2 + Ry2
To find angle or direction of resultant, tangent of angle vector makes with x-axis is:
Angle of Resultant Vector

12. Vectors Vectors in Multiple Dimensions
Jeff moved 3 m due north, and then 4 m due west to his friend’s house. What is the displacement of Jeff?

13. Vectors Vectors in Multiple Dimensions
Calculate the resultant of the three vectors A, B, and C as shown in the figure.
(Ax = Bx = Cx = Ay = Cy = 1 units and By = 2 units)

14. Vectors Vectors in Multiple Dimensions
If a vector B is resolved into two components Bx and By, and if  is the angle that vector B makes with the positive direction of x-axis, which of the following formulae can you use to calculate the components of vector B?

15. Vectors Vectors in Multiple Dimensions
A small plane takes off and flies 12.0 km in a direction southeast of the airport. At this point the plane turns 20.0° to the east of its original flight path and flies 21.0 km. What is the magnitude of the plane’s resultant displacement from the airport?
A hammer slides down a roof that makes a 32.0° angle with the horizontal. What are the magnitudes of the components of the hammer’s velocity at the edge of the roof if it is moving at a speed of m/s

16. Friction
Friction
When you push your hand across a surface you feel the force called friction opposing the motion of your hand
Friction is a force that resists motion whenever the surfaces of two objects rub against each other
There are two types of friction, and both always oppose the initial motion of an object sliding across a surface

17. Friction Kinetic Friction
When you push a book across the desk, it experiences a type of friction that acts on moving objects
This force is known as kinetic friction, and it is exerted on one surface by another when the two surfaces rub against each other because one or both of them are moving

18. Friction Static Friction
To understand the other kind of friction, imagine trying to push a heavy couch across the floor. You give it a push, but it does not move
Because it does not move, Newton’s laws tell you that there must be a second horizontal force acting on the couch, one that opposes your force and is equal in size
This force is static friction, which is the force exerted on one surface by another when there is no motion between the two surfaces

19. Friction Static Friction
You might push harder and harder, as shown in the figure below, but if the couch still does not move, the force of friction must be getting larger
This is because the static friction force acts in response to other forces. Finally, when you push hard enough, as shown in the figure below, the couch will begin to move

20. Friction Static and Kinetic Friction
There is a limit to how large the static friction force can be. Once your force is greater than this maximum static friction, the couch begins moving and kinetic friction begins to act on it instead of static friction
Frictional force depends on the materials that the surfaces are made of
For example, there is more friction between skis and concrete than there is between skis and snow
The normal force (FN) between the two objects also matters. The harder one object is pushed against the other, the greater the force of friction that results.

21. Friction Static and Kinetic Friction
If you pull a block along a surface at a constant velocity, according to Newton’s laws, the frictional force must be equal and opposite to the force with which you pull
You can pull a block of known mass along a table at a constant velocity and use a spring scale, as shown in the figure, to measure the force that you exert

22. Friction Static and Kinetic Friction
You can then stack additional blocks on the block to increase the normal force and repeat the measurement
Plotting the data will yield a graph like the one shown here. There is a direct proportion between the kinetic friction force and the normal force

23. Friction Static and Kinetic Friction
The different lines correspond to dragging the block along different surfaces
Note that the line corresponding to the sandpaper surface has a steeper slope than the line for the highly polished table
You would expect it to be much harder
to pull the block along sandpaper than
along a polished table, so the slope must
be related to the magnitude of the
resulting frictional force

24. Friction Kinetic Friction Force Static and Kinetic Friction
The slope of this line, designated μk, is called the coefficient of kinetic friction between the two surfaces and relates the frictional force to the normal force, as shown below
The kinetic friction force is equal to the product of the coefficient of the kinetic friction and the normal force
Kinetic Friction Force

25. Friction Static and Kinetic Friction
The maximum static friction force is related to the normal force in a similar way as the kinetic friction force
The static friction force acts in response to a force trying to cause a stationary object to start moving. If there is no such force acting on an object, the static friction force is zero
If there is a force trying to cause motion, the static friction force will increase up to a maximum value before it is overcome and motion starts

26. Friction Static Friction Force Static and Kinetic Friction
The static friction force is less than or equal to the product of the coefficient of the static friction and the normal force
In the equation for the maximum static friction force, μs is the coefficient of static friction between the two surfaces, and μsFN is the maximum static friction force that must be overcome before motion can begin
Static Friction Force

27. Friction Static and Kinetic Friction
Note that the equations for the kinetic and maximum static friction forces involve only the magnitudes of the forces
The forces themselves, Ff and FN, are at right angles to each other. The table here shows coefficients of friction between various surfaces
Although all the listed coefficients are less than 1.0, this does not mean that they must always be less than 1.0

28. Friction Static and Kinetic Friction
You push a 25.0 kg wooden box across a wooden floor at a constant speed of 1.0 m/s. How much force do you exert on the box (μk = 0.20) ?

29. Friction Static and Kinetic Friction
A force of 110 N is required to move a table of 120 kg. What is the coefficient of static friction (μs) between the table and the floor?
A worker has to move a 17.0 kg crate along a flat floor in a warehouse. The coefficient of kinetic friction between the crate and the floor is The worker pulls horizontally on a rope attached to the crate, with a 49.0 N force. What is the resultant acceleration of the crate?

30. Force in Two Dimensions
Equilibrium Revisited
When net force on an object is zero, the object is in equilibrium
According to Newton’s laws, the object will not accelerate with no net force acting on it
Object in equilibrium is motionless or moves with constant velocity

31. Force in Two Dimensions
Equilibrium Revisited
Equilibrium can occur no matter how many forces act on an object
If resultant zero, net force zero and object in equilibrium
Figure shows three forces exerted on a point object. What is the net force acting on the object?
Vectors may be moved if direction
(angle) or length is not changed

32. Force in Two Dimensions
Equilibrium Revisited
Figure shows addition of three forces, A, B, and C
Vectors form closed triangle
No net force; thus, sum of the forces is zero and object in equilibrium

33. Force in Two Dimensions
Equilibrium Revisited
Suppose two forces are exerted on object with sum not zero
How could you find third force that adds to other two totals zero, and causes object to be in equilibrium?
First find sum of two forces already exerted on object
Single force that produces same effect as two summed individual forces, is resultant force
Force to find has same magnitude as resultant force, but in opposite direction
Equilibrant – Force that puts object in equilibrium

34. Force in Two Dimensions
Equilibrium Revisited
The figure below illustrates the procedure for finding the equilibrant for two vectors

35. Force in Two Dimensions
Motion Along an Inclined Plane

36. Force in Two Dimensions
Motion Along an Inclined Plane
Because an object’s acceleration is usually parallel to the slope, one axis, usually the x-axis, should be in that direction
The y-axis is perpendicular to the x-axis and perpendicular to the surface of the slope
With this coordinate system, there are two forces—normal and frictional forces. These forces are in the direction of the coordinate axes. However, the weight is not

37. Force in Two Dimensions
Motion Along an Inclined Plane
This means that when an object is placed on an inclined plane, the magnitude of the normal force between the object and the plane will usually not be equal to the object’s weight
You will need to apply Newton’s laws once in the x-direction and once in the y-direction
Because the weight does not point in either of these directions, you will need to break this vector into its x- and y-components before you can sum your forces in these two directions

38. Force in Two Dimensions
Motion Along an Inclined Plane
A child shoves a small toboggan weighing N up a snowy hill, giving the toboggan an initial speed of 6.0 m/s. If the hill is inclined at an angle of 32.0° above the horizontal, how far along the hill will the toboggan slide? Assume that the coefficient of sliding friction between the toboggan and the snow is 0.15.