Juan is sitting on a sled on the side of a hill inclined at ,with Rafaela holding Juan from sliding down the hill. Thecombined weight of Juan and the sled is 140 lb. What force isrequired for Rafaela to keep the sled from sliding down the hill?

45

Applications ofApplications ofVector ProjectionVector Projection

6.2b6.2b

Consider the situation of pulling a box along the ground witha rope that is not parallel to the ground:

Obviously, the box will move in the direction of v

v

u

If I pull with force u, then the effective force acting on the box is in the direction of v, the vector projection of u onto v.

proj uv

F = 70, 70 , |F| = 70 2 = 98.995 lb. Remember this example when you do #45 and 46 on the HW!!!

Juan is sitting on a sled on the side of a hill inclined at 45 ,with Rafaela holding Juan from sliding down the hill. Thecombined weight of Juan and the sled is 140 lb. What force isrequired for Rafaela to keep the sled from sliding down the hill?

140F j2 2

cos( ) sin( )4 4 2 2

v i j i j

FF1

21 ( ) ( )v

F vF proj F v F v v

v

Proctor is once again sitting on an incline, this time with a29 angle of inclination. If Proctor weighs 199 lb, what is theforce required to keep Proctor from sliding down the hill?What is the component of Proctor’s weight that is perp. tothe hill? (use vector projection to solve)

|F| = 96.477 lb, |F | = 174.049 lb

A 120 pound force F is pulling a box up an inclined plane. IfF makes an angle of 15 with the plane, and the plane makesan angle of 21 with the horizontal, find the magnitude of theeffective force pulling the box up the plane.

15

21

120 lb

F = 108.212, 41.539 , |F | = 115.911 lb.R R

From physics-land…In general, work is defined as a force applied over a distance

Do you think work is a scalar or a vector???Do you think work is a scalar or a vector???

In vector notation, if a force F is acting in the same directionof AB, then the work (W) done by F in moving an object fromA to B is

W = |F| |AB|

From physics-land…

If a force F is acting in any direction, then the work (W) doneby F in moving an object from A to B is

W = F AB

But what if the force is not acting inBut what if the force is not acting inthe same direction as the motion?...the same direction as the motion?...

= |F| |AB| cos 0

where 0 is the angle between F and AB.

Practice Problems

Find the work done by a 10 pound force acting in the direction 1, 2 in moving an object 3 feet from (0, 0) to (3, 0).

W = 13.416 foot-poundsW = 13.416 foot-pounds

Let’s solve this one using both versions of the formula…Let’s solve this one using both versions of the formula…

First, a diagram…W F AB!!!!!!!!!!!!!!

Component form of each vector:

AB 3,0!!!!!!!!!!!!!!

1,2F 10

1,2

10 20,5 5

10 20W , 3,0

5 5

13.416

Practice Problems

Find the work done by a 10 pound force acting in the direction 1, 2 in moving an object 3 feet from (0, 0) to (3, 0).

W = 13.416 foot-poundsW = 13.416 foot-pounds

Let’s solve this one using both versions of the formula…Let’s solve this one using both versions of the formula…

Angle between the vectors:

W F AB cos!!!!!!!!!!!!!!

Magnitude of each vector:

AB 3!!!!!!!!!!!!!!

F 10 W 10 3 cos63.435 13.416

1tan 2 63.435

Practice Problems

Find the work done by a force F of 24 Newtons acting in thedirection 4, 5 in moving an object 5 meters from (0, 0) to(5, 0).

W = 74.963 N-mW = 74.963 N-m

Practice Problems

Find the work done by lifting a 199 pound math teacher 15 feet.

W = 2985 ft-lbW = 2985 ft-lb

Practice Problems

Find the work done by a force F of 50 pounds acting in thedirection 2, 3 in moving an object 5 feet from (0, 0) to apoint in the first quadrant along the line y = x.

W = 245.145 ft-lbW = 245.145 ft-lb

Component form of each vector:

AB 2.5 2,2.5 2!!!!!!!!!!!!!! 2,3

F 502,3

100 150

,13 13

First, the diagram…

100 150W , 2.5 2,2.5 2

13 13 245.145

Practice ProblemsThe angle between a 75 pound force F and AB is 60 , whereA = (–1, 1) and B = (4, 3). Find the work done by F in movingan object from A to B.

W = 201.944 ft-lbW = 201.944 ft-lb

AB 4 1,3 1 !!!!!!!!!!!!!!

5,2

2 2AB 5 2 !!!!!!!!!!!!!!

F 7529

W F AB cos!!!!!!!!!!!!!!

201.94475 29 cos60