1. Relative Equilibrium of
Liquids

2. • Horizontal Motion
• Inclined Motion
• Vertical Motion

3. Horizontal Motion
𝐭𝐚𝐧 𝜽 =
𝒂
𝒈

4. Inclined Motion
𝐭𝐚𝐧 𝜽 =
𝒂𝒉
𝒈 ± 𝒂𝒗
Use (+) sign for upward
motion and (-) sign for
downward motion.

5. Vertical Motion
𝒑 = 𝜸𝒉(𝟏 ±
𝒂
𝒈
)
Use (+) for upward motion
Use (-) for downward motion
Also note that a is positive for
acceleration and negative for
deceleration.

6. Example:
1. An open horizontal tank 1.5m high, 2m wide and 5m
long is full of water. It is accelerated horizontally in the
direction parallel to the length of the tank. The
magnitude of the acceleration is 2 m/s2.
a. How much water is spilled out?
b. What is the force acting on the side with the greatest
depth?
c. What is the force acting on the side with the least
depth?

7. A gasoline tanker 2m wide, 10m long and 2m deep
contains gasoline (s=0.75) to a depth of 1.2m.
a. Determine the maximum horizontal acceleration
of the tanker such that the gasoline will just reach
its top end.
b. If the tanker is closed and completely filled with
the gasoline and accelerated horizontally at 2m/s2,
determine the total liquid thrust on the front end.
c. Determine the liquid thrust on the rear end.

8. A drum containing water to a depth to a depth of
1.2m is being raised upward and downward.
a. Find the pressure at the bottom of the drum if
the velocity is constant.
b. Find the pressure at the bottom of the drum if it
is accelerating upward at the rate of 4m/s2.
c. Find the pressure at the bottom of the drum if it
is accelerating downward at a rate 3m/s2.

9. The open railcar is used to transport water up the
20° incline. When the car is at rest, the water level
is as shown.
a. Determine the maximum acceleration the car
can have when it is pulled up the incline so that no
water will spill out.
b. Determine the maximum acceleration the car
can have when it is pushed down the incline so
that no water will spill out.

10. Rotation - Rotating Vessel
𝐭𝐚𝐧 𝜽 =
𝝎𝟐
𝒙
𝒈
𝒚 =
𝒘𝟐
𝒙𝟐
𝟐𝒈
For cylindrical vessel of
radius r revolved about its
vertical axis, the height h of
paraboloid is
𝒉 =
𝒘𝟐
𝒓𝟐
𝟐𝒈

11. Other Formulas
• By squared-property of
parabola, the relationship
of y, x, h and r is defined
by
𝒓𝟐
𝒉
=
𝒙𝟐
𝒚
Volume of paraboloid of revolution
𝑽 =
𝟏
𝟐
𝝅𝒓𝟐
𝒉
Important conversion factor
𝟏 𝒓𝒑𝒎 =
𝟏
𝟑𝟎
𝝅
𝒓𝒂𝒅
𝒔𝒆𝒄

12. An open cylindrical tank 0.3 m in diameter and
0.80 m high is partially filled with water to a certain
depth. It is then rotated about its vertical axis at
240 rpm but no water spilled out.
a. How deep is the water in the tank?
b. At what speed in rpm, would the tank be rotated
if 1.4 liters of water is spilled out?
c. At what speed in rpm, would the tank be rotated
so that the pressure at the center of the bottom of
the tank be zero?

13. A closed cylindrical tank having a diameter of 2m
and a height of 5m is filled with water and rotated
about its own vertical axis.
a. Determine the maximum pressure at the bottom
of the tank if it is rotated at 120 rpm.
b. Determine the minimum pressure at the bottom
of the tank.
c. If the cylinder is 3mm thick, determine the
maximum rpm so that the tank is at the verge of
bursting. The ultimate strength of the tank is 248
MPa.

14. A closed cylindrical vessel 3 m. in diameter and 6
m high is filled with water to a height of 4.5 m. The
rest is filled with air, the pressure of which is 105
kPa. If the vessel is rotated at 191 rpm about its
axis, determine the maximum and minimum inside
pressure at the base.