1. 8.3 – Applications to Physics and Engineering
Math 181
8.3 – Applications to Physics and Engineering

2. Hydrostatic Pressure and Force

3. The deeper you go underwater, the more the pressure. Why
The deeper you go underwater, the more the pressure. Why? Because you have a taller and taller column of water above you, pushing down on you.

4. If we place a thin plate of area 𝐴 horizontally underwater at a depth of 𝑑, then the volume of the column of water above it would be 𝑉=𝐴𝑑.

5. The mass of the water would be 𝑚=𝜌𝑉=𝜌𝐴𝑑 (where 𝜌 is the mass-density of water). So, the force of the water acting on the plate would be 𝐹=𝑚𝑔=𝜌𝑔𝐴𝑑.

6. The mass of the water would be 𝑚=𝜌𝑉=𝜌𝐴𝑑 (where 𝜌 is the mass-density of water). So, the force of the water acting on the plate would be 𝐹=𝑚𝑔=𝜌𝑔𝐴𝑑.

7. Note that pressure (force per unit area) is 𝑃= 𝐹 𝐴 = 𝜌𝑔𝐴𝑑 𝐴 =𝜌𝑔𝑑
Note that pressure (force per unit area) is 𝑃= 𝐹 𝐴 = 𝜌𝑔𝐴𝑑 𝐴 =𝜌𝑔𝑑. So, you can also find force via pressure by the equation 𝐹=𝑃𝐴.

8. Note that pressure (force per unit area) is 𝑃= 𝐹 𝐴 = 𝜌𝑔𝐴𝑑 𝐴 =𝜌𝑔𝑑
Note that pressure (force per unit area) is 𝑃= 𝐹 𝐴 = 𝜌𝑔𝐴𝑑 𝐴 =𝜌𝑔𝑑. So, you can also find force via pressure by the equation 𝐹=𝑃𝐴.

9. Notes:
At any point of a given depth, the pressure applied in all directions is the same.
𝑃=𝜌𝑔𝑑 shows that this pressure varies linearly with depth.
For water, we’ll use 𝜌=1000 kg/ m 3 and 𝜌𝑔=62.5 lb/f t 3 .
We can use the concept of a thin, flat plate to represent a surface (like the side of a swimming pool, the side of a trough, or the side of a dam).

10. Notes:
At any point of a given depth, the pressure applied in all directions is the same.
𝑃=𝜌𝑔𝑑 shows that this pressure varies linearly with depth.
For water, we’ll use 𝜌=1000 kg/ m 3 and 𝜌𝑔=62.5 lb/f t 3 .
We can use the concept of a thin, flat plate to represent a surface (like the side of a swimming pool, the side of a trough, or the side of a dam).

11. Notes:
At any point of a given depth, the pressure applied in all directions is the same.
𝑃=𝜌𝑔𝑑 shows that this pressure varies linearly with depth.
For water, we’ll use 𝜌=1000 kg/ m 3 and 𝜌𝑔=62.5 lb/f t 3 .
We can use the concept of a thin, flat plate to represent a surface (like the side of a swimming pool, the side of a trough, or the side of a dam).

12. Notes:
At any point of a given depth, the pressure applied in all directions is the same.
𝑃=𝜌𝑔𝑑 shows that this pressure varies linearly with depth.
For water, we’ll use 𝜌=1000 kg/ m 3 and 𝜌𝑔=62.5 lb/f t 3 .
We can use the concept of a thin, flat plate to represent a surface (like the side of a swimming pool, the side of a trough, or the side of a dam).

13. Ex 1. A swimming pool full of water is 12 ft wide, 20 ft long, and 5 ft deep. What is the hydrostatic force acting on the bottom of the pool?

14. What if the plate is vertical instead
What if the plate is vertical instead? Then the depth is not constant anymore since it varies from the bottom to the top of the plate. Ex 2. What is the hydrostatic force acting on a 12-ft-by-5-ft side of the pool described in Ex 1?

15. What if the plate is vertical instead
What if the plate is vertical instead? Then the depth is not constant anymore since it varies from the bottom to the top of the plate. Ex 2. What is the hydrostatic force acting on a 12-ft-by-5-ft side of the pool described in Ex 1?

16. Ex 3. A vertical plate is partially submerged in water as shown
Ex 3. A vertical plate is partially submerged in water as shown. Use a Riemann sum to approximate the hydrostatic force against one side of the plate. Then find the exact hydrostatic force against one side of the plate.

17. Ex 4. Same directions as above.

18. Center of Mass: One-dimension

19. _______ is the “turning effect” of a mass on a rigid body with a fulcrum. The above system has torque: _________________________ (Note: 𝑥 1 , 𝑥 2 , and 𝑥 3 are signed.)

20. _______ is the “turning effect” of a mass on a rigid body with a fulcrum. The above system has torque: _________________________ (Note: 𝑥 1 , 𝑥 2 , and 𝑥 3 are signed.)
Torque

21. _______ is the “turning effect” of a mass on a rigid body with a fulcrum. The above system has torque: _________________________ (Note: 𝑥 1 , 𝑥 2 , and 𝑥 3 are signed.)
Torque

22. _______ is the “turning effect” of a mass on a rigid body with a fulcrum. The above system has torque: _________________________ (Note: 𝑥 1 , 𝑥 2 , and 𝑥 3 are signed.)
Torque
𝒎 𝟏 𝒈 𝒙 𝟏 + 𝒎 𝟐 𝒈 𝒙 𝟐 + 𝒎 𝟑 𝒈 𝒙 𝟑

23. We can change the torque of the system by moving the fulcrum
We can change the torque of the system by moving the fulcrum. The torque is 0 when the fulcrum is placed at the ______________ (labeled 𝑥 ). How can we calculate this 𝑥-value?

24. We can change the torque of the system by moving the fulcrum
We can change the torque of the system by moving the fulcrum. The torque is 0 when the fulcrum is placed at the ______________ (labeled 𝑥 ). How can we calculate this 𝑥-value?
center of mass

25. We can change the torque of the system by moving the fulcrum
We can change the torque of the system by moving the fulcrum. The torque is 0 when the fulcrum is placed at the ______________ (labeled 𝑥 ). How can we calculate this 𝑥-value?
center of mass

26. Note: ∑ 𝑚 𝑖 𝑥 𝑖 is called the _________ of the system about the origin (notice that it’s just the torque of the system about the origin with acceleration removed). Note: 𝑀=∑ 𝑚 𝑖 𝑥 𝑖 and 𝑚=∑ 𝑚 𝑖

27. Note: ∑ 𝑚 𝑖 𝑥 𝑖 is called the _________ of the system about the origin (notice that it’s just the torque of the system about the origin with acceleration removed). Note: 𝑀=∑ 𝑚 𝑖 𝑥 𝑖 and 𝑚=∑ 𝑚 𝑖
moment

28. Note: ∑ 𝑚 𝑖 𝑥 𝑖 is called the _________ of the system about the origin (notice that it’s just the torque of the system about the origin with acceleration removed). Note: 𝑀=∑ 𝑚 𝑖 𝑥 𝑖 and 𝑚=∑ 𝑚 𝑖
moment

29. Center of Mass: Two-dimensions

30. With masses on a plane (two-dimensions), our center of mass will have two coordinates: ( 𝑥 , 𝑦 ) 𝑥 = ∑ 𝑚 𝑖 𝑥 𝑖 ∑ 𝑚 𝑖 = 𝑀 𝑥 𝑚 𝑦 = ∑ 𝑚 𝑖 𝑦 𝑖 ∑ 𝑚 𝑖 = 𝑀 𝑦 𝑚

36. For a thin plate (lamina) that lies in a plane (like a circular or triangular sheet of metal), we can imagine cutting the plate into strips.

37. The center of mass of all the strips is again ( 𝑥 , 𝑦 ), but here 𝑥 = ∑ 𝑥 𝑖 Δ 𝑚 𝑖 ∑Δ 𝑚 𝑖 𝑦 = ∑ 𝑦 𝑖 Δ 𝑚 𝑖 ∑Δ 𝑚 𝑖 With more and thinner strips, the finite sums become integrals, and we get the following formulas: 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 𝑦 = ∫ 𝑦 𝑑𝑚 ∫𝑑𝑚

38. The center of mass of all the strips is again ( 𝑥 , 𝑦 ), but here 𝑥 = ∑ 𝑥 𝑖 Δ 𝑚 𝑖 ∑Δ 𝑚 𝑖 𝑦 = ∑ 𝑦 𝑖 Δ 𝑚 𝑖 ∑Δ 𝑚 𝑖 With more and thinner strips, the finite sums become integrals, and we get the following formulas: 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 𝑦 = ∫ 𝑦 𝑑𝑚 ∫𝑑𝑚

39. 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 𝑦 = ∫ 𝑦 𝑑𝑚 ∫𝑑𝑚 Note: 𝑑𝑚 represents the mass of a strip, and can be found by multiplying the area density (𝜌) by the area of the strip (𝑑𝐴).

40. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.

41. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.

42. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.

43. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.

44. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.

45. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.
centroid

46. When the area density 𝜌 is uniform/constant, we have 𝑥 = ∫ 𝑥 𝑑𝑚 ∫𝑑𝑚 = ∫ 𝑥 𝜌𝑑𝐴 ∫𝜌𝑑𝐴 = 𝜌∫ 𝑥 𝑑𝐴 𝜌∫𝑑𝐴 = ∫ 𝑥 𝑑𝐴 ∫𝑑𝐴 In this case, the center of mass is also called the _________. Note: 𝑑𝐴 just represents the area of the plate/region/lamina, which we can just call 𝐴.
centroid

47. So, the centroid of a plate/region/lamina has the coordinates: 𝒙 = 𝟏 𝑨 𝒙 𝒅𝑨 𝒚 = 𝟏 𝑨 𝒚 𝒅𝑨

48. Ex 5. Find the centroid of the rectangular plate shown to the right.

49. Ex 5. Find the centroid of the rectangular plate shown to the right.

50. Ex 5. Find the centroid of the rectangular plate shown to the right.

51. Ex 5. Find the centroid of the rectangular plate shown to the right.

52. Ex 6. Find the centroid of the triangular plate shown to the right.

53. Ex 6. Find the centroid of the triangular plate shown to the right.

54. Ex 6. Find the centroid of the triangular plate shown to the right.

55. Ex 6. Find the centroid of the triangular plate shown to the right.

56. Ex 6. Find the centroid of the triangular plate shown to the right.

57. Ex 6. Find the centroid of the triangular plate shown to the right.

58. Ex 7. Find the centroid of the region enclosed by 𝑦=𝑥 and 𝑦= 𝑥 2 .

59. Ex 8. Find the centroid of the region by 𝑥= 𝑦 2 −𝑦 and 𝑥=𝑦.