1. Friction
Additional Examples

2. Example 1: A 100-lb force acts as shown on a 300-lb block placed on an inclined plane. The coefficients of friction between the block and the plane are ms =0.25 and mk = Determine whether the block is in equilibrium, and find the value of the friction force.

3. Force required for equilibrium: Assuming that F is directed down
SFx = 100lb – 3/5 (300lb) – F = 0
F = 80 lb
SFy = N – 4/5(300lb) = 0, N = +240 lb
The F required to maintain equilibrium is directed up and to the right; the tendency of the block is move down the plane.
2. Maximum friction force.
Fm = ms N, Fm = 0.25(240 lb) = 60 lb
Since the value of F required to maintain equilibrium (80lb) is larger than the maximum possible (60lb), the block will slide down the plane

4. 3. Actual value of friction force:
Factual = Fk ( the body is moving)
Factual = Fk = mk N = 0.2(240lb) = 48 lb
The sense of this force is opposite to the sense of motion.
The forces acting on the block are not balanced, the resultant is:
3/5 (300lb) – 100lb – 48lb = 32lb

5. Example2: Determine whether the block shown is in equilibrium and find the magnitude and direction of the friction force when q = 30o and P = 50lb.

6. y x q q N F Assume equilibrium: SFy =N – 250cos30o-50sin30o = 0
N = lb
SFx =F– 250sin30o +50cos30o = 0, F = lb x q q N F
2. Maximum friction force:
Fm = ms N = 0.3 (241.5 lb) = 72.5 lb
Since F > Fm, then the block moves down
Friction force: F = mk N = 0.2(241.5lb) = 48.3 lb

7. Example 3: The block in the figure has a mass of 100 kg
Example 3: The block in the figure has a mass of 100 kg. The coefficient of friction between the block and the inclined surface is 0.2. (a) Determine if the system is in equilibrium when P= 600 N P
20o
30o

8. W= 981 N y P x 30o 20o F N 30o Determination of F and N:
SFy= Psin 20o + N – Wcos 30o = 0, N = N
SFx= Pcos 20o – F – Wsin 30o = 0, F = N
2. Determination of Fmaximum
Fm =msN= (0.20)(644.36)=128.87N
Since Fm > 73.32, then
the block is in equilibrium. P x
30o
20o F N
30o

9. b) Determine the minimum force P to prevent motiony
Pmin
The minimum P will be required when motion of the block down the incline is impending.
F must resist this motion as shown.
Equilibrium exists when:
SFx = Pmin cos 20o + F – 981sin 30o = 0
SFx = Pmin cos 20o + 0.2N – 981 sin 30o = 0
SFy = Pmin sin 20o + N – 981cos 30o = 0
Then N = 724 N, P min = 368 N x
30o
20o F N
30o

10. c) Determine the maximum force P for which the system is in equilibrium
The maximum force P will be required when motion of the block up the incline is impending.
For this condition, F will tend to resist this motion as shown. Then:
SFx = Pmax cos 20o – 0.2 N – 981 sin 30o = 0
SFy = Pmax sin 20o + N – 981 cos 30o = 0
Solving simultaneously:
N = 626 N
Pmax = 655N
981 N y
Pmax x
30o
20o F N
30o

11. Center of Gravity
Additional Examples

12. Centroids – Simple Example for a Composite Body
Find the centroid of the given body

13. Centroids – Simple Example for a Composite Body
To find the centroid,
Determine the area of the components

14. Centroids – Simple Example for a Composite Body
The total area is

15. Centroids – Simple Example for a Composite Body
To centroid of each component
Compute the x centroid

16. Centroids – Simple Example for a Composite Body
To centroid of each component
Compute the y centroid

17. Centroids – Simple Example for a Composite Body
The problem can be done using a table to represent the composite body.

18. Centroids – Simple Example for a Composite Body
An alternative method of computing the centroid is to subtract areas from a total area.
Assume that area is
Assume that area is a large square and subtract the small triangular area.

19. Centroids – Simple Example for a Composite Body
The problem can be done using a table to represent the composite body.

20. Centroids –Example for a Composite Body
Find the centroid of the given body

21. Centroids –Example for a Composite Body
Determine the area of the components

22. Centroids –Example for a Composite Body
The total area is

23. Centroids – Example for a Composite Body

24. Centroids – Example for a Composite Body
An Alternative Method would be to subtract to areas

25. Centroids – Class Problem
Find the centroid of the body

26. Centroids – Class Problem
Find the centroid of the body