Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
An open rectangular tank is fixed on a truck. The tank is long (along the road), wide, and its walls are high. With the truck at rest, it holds water to a depth of , as shown in the figure. The truck then moves off along a straight, level road with a steady acceleration of , and the water settles into a new shape (it moves with the tank). Take and the density of water as . Give pressures as gauge pressures (above the air's pressure).
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. Work in the truck's frame, where the water is at rest under its weight and a pseudo force. Find the tilt of the surface, then use the depth at each wall.
Step 1. The tilt (a). Each bit of water of mass feels downwards and backwards. The free surface is at right angles to the combined pull, so it makes an angle with the horizontal where
It slopes up towards the rear of the truck.
Step 2. Depths (b). The amount of water is fixed, so the surface turns about the middle of the tank. Over half the length, , it rises by . So the depth is at the rear and at the front.
Step 3. Pressures (c). Vertically nothing has changed: the pressure grows by per metre of depth below the surface above that point. At the rear, ; at the front, .
Step 4. Forces on the end walls (d). On a vertical wall wetted to a depth , the pressure grows steadily from zero at the top to at the bottom, so the force is times the width:
Step 5. The limit (e). Water spills when the rear depth reaches the top of the wall, a rise of over half the length: , so .
Answer: (a) . (b) at the rear, at the front. (c) and . (d) on the rear wall, on the front wall. (e) .
Check. The tank's walls must push the water forwards to give it its acceleration. The water's mass is , so the net push needed is . The end walls give . They agree.
Watch out. The pressure at the middle of the bottom stays the same as at rest, . The pressure grows towards the rear and falls towards the front, because the depth changes, not because the rule fails.
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