# Calculating torque for a small horizontal strip

• Swatch
In summary, the net torque on a gate due to the water force is not the right answer, which is 2.61*10^4 N*m.
Swatch
The uppper edge af a gate in a dam runs along the water surface. The gate is 2.00 m high and 4.00 m wide and is hinged along a horizontal line trough its center. I have to calculate the torque about the hinge arising from the force due to the water. What I have done so far is to consider the top of the gate as point y=0 , the center as y=1 and the bottom y=2.
Then I calculate the torque of a thin horizontal strip at a depth y and integrate that over the gate.

Total torque from the top to the center would be

Torque=intergrate(r*g* dy *dA ) or 4*r*g*integrate(y^3)
(where r=density of water or 1.00*10^3 kg/m^3)

from this I get torque to be 9800 N*m

Then I do the same for the gate from height y=1 to y=2
and get -147000

The net torque I get is not the right answer, which is by the way2.61*10^4 N*m

Could someone please give me a hint to what I am doing wrong

It's not clear to me what you are integrating. The force on a thin horizontal strip is $dF = 4 \rho g y dy$. For the top half of the gate, the torque on that piece (about the center) is $d\tau = (1-y) 4 \rho g y dy$. Now integrate from y = 0 to 1 to find the net torque on the top half.

Write a similar expression for the torque on the bottom half.

Did what you told me Doc Al, and got the right answer. Just one thing I'm not understanding. You say that the force on a thin horizontal strip is dF=4*rho*g dy

The way I see the force is dF=dp*dA
since dp=rho*g dy and dA=4*dy
shouldn't the integration contain y^2

Swatch said:
Just one thing I'm not understanding. You say that the force on a thin horizontal strip is dF=4*rho*g dy
No, I said: $dF = 4 \rho g y dy$ (Don't forget the y.)

The way I see the force is dF=dp*dA
since dp=rho*g dy and dA=4*dy
shouldn't the integration contain y^2
$dF = P dA$, not $dP dA$. You need the actual pressure at the depth y: $P = \rho g y$, not $\rho g dy$.

O.K.

So at this little horizontal strip or dA I consider the pressure as constant. Is that the correct way to think of it?

Swatch said:
So at this little horizontal strip or dA I consider the pressure as constant. Is that the correct way to think of it?
Right. For a small horizontal strip you can consider the pressure to be uniform, just like you can consider it to be at a single depth.

(FYI: Realize that the pressure varies from P to P + dP across the thickness of the strip. So, the average pressure is P + dP/2. So the average force would be (P + dp/2)(dA) = PdA + dp dA/2. Speaking loosely, that last term is a higher order infinitesimal and can be ignored compared to PdA.)

## 1. What is a dam?

A dam is a man-made structure built across a river or other body of water to prevent or control the flow of water. It is typically used to create a reservoir for water storage, generate hydroelectric power, or control flooding.

## 2. How does a dam work?

A dam works by blocking the natural flow of water and creating a barrier. This causes water to accumulate behind the dam, creating a reservoir. The water can then be released in a controlled manner to regulate the flow downstream, generate electricity, or store water for future use.

## 3. What is torque?

Torque is a measure of a force that can cause an object to rotate or turn. It is typically measured in units of foot-pounds (ft-lb) or newton-meters (N-m) and is dependent on both the magnitude and direction of the force.

## 4. How does torque relate to a dam?

In a dam, torque is an important factor in determining the structural stability of the dam. The weight of the water in the reservoir exerts a force on the dam, which can cause it to rotate or turn. Engineers must consider torque when designing a dam to ensure it can withstand the force of the water without collapsing.

## 5. What are some common torque problems in dams?

Some common torque problems in dams include excessive rotation or movement of the dam, leakage or seepage of water under or through the dam, and erosion or shifting of the foundation. These issues can compromise the structural integrity of the dam and must be closely monitored and addressed to prevent failures.

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