Problem Statement
A sliding gate 2 m wide and 1.5 m high lies in a vertical plane and has a co-efficient of friction of 0.2 between itself and guides. If the gate weighs one tonne, find the vertical force required to raise the gate if its upper edge is at a depth of 4 m from the free surface of water.
Given Data
- Gate width, \( b = 2 \, \text{m} \)
- Gate height, \( d = 1.5 \, \text{m} \)
- Coefficient of friction, \( \mu = 0.2 \)
- Mass of gate = 1 tonne = \( 1000 \, \text{kg} \)
- Depth of upper edge = 4 m
Solution
To find the vertical force required to raise the gate, we must overcome its weight and the frictional force acting against the motion. The frictional force depends on the hydrostatic thrust from the water.
1. Hydrostatic Thrust on the Gate (\(F_H\))
First, calculate the area of the gate and the depth of its centroid.
The hydrostatic thrust is the force exerted by the water, which acts as the normal force for friction.
2. Frictional Force (\(F_f\))
The frictional force opposes the upward motion and is calculated using the coefficient of friction and the hydrostatic thrust.
3. Total Vertical Force Required (\(F_V\))
The total force required to raise the gate is the sum of the gate's weight and the downward frictional force.
The vertical force required to raise the gate is \( F_V \approx 37.82 \, \text{kN} \).
Explanation of Concepts
Hydrostatic Thrust as Normal Force: The horizontal force exerted by the water on the face of the gate (the hydrostatic thrust) pushes the gate against its guides. This thrust acts as the normal force in the friction calculation.
Frictional Force: Friction is a force that resists motion between surfaces in contact. It is calculated as the product of the coefficient of friction (\(\mu\)) and the normal force. Since we want to raise the gate, the motion is upward, and the frictional force acts in the opposite (downward) direction.
Force Equilibrium: To just begin to raise the gate, the upward applied force (\(F_V\)) must be equal to the sum of all downward forces. In this case, the downward forces are the gate's own weight and the frictional force from the guides.
