Physics For Architects
Many interesting topics were not included in "Physics for Architects", because of their low relevancy to architecture. Here are two of them: (No prior knowledge of physics or sailing is required.)
HOW SAILBOATS SAIL AGAINST THE WIND
IN A NUTSHELL
The direction in which
a sailboat sails depends on the force of the wind and on the resistance
of the water. For certain orientations of the boat and its sails with
respect to the wind, the combined effect of the wind and the water is
a net force that pushes the boat diagonally into the wind.
The force that the wind exerts on the sail has two components: The drag component pushes the sail in the direction of the wind, and the lift component pushes the sail perpendicular to the wind. Due to the lift, the direction of the total wind-force is different from the direction in which the wind is blowing (the direction of the wind). The direction of the wind-force depends on the shape of the sail and on the angle between the sail and the wind.
The resistance of the water slows down the boats forward motion and her sidewise slippage. For a boat to be able to sail diagonally into the wind, her sidewise slippage has to be very small compared to her forward motion. A keel significantly reduces sidewise slippage, while having a small effect on the forward motion.
If the keel practically eliminates sidewise slippage, the boat can move only in the direction of the keel, which is also the direction of her center line. Whenever the total wind-force points diagonally forwards with respect to keel, the boat moves forwards, in the direction of her keel. If the keel points diagonally into the wind and the wind-force points diagonally forwards with respect to the keel, the boat will sail diagonally into the wind. If the sidewise slippage is too big, the boat wont be able to sail diagonally into the wind.
The keel and the rudder
Viscosity force acts on objects that move in liquid. It opposes their motion. A narrow object encounters less viscosity resistance than a wide one. That is why it is easier for a boat to move in her long direction than to move sidewise. However, strong winds may push a boat sidewise. Keels increase the resistance of the boat to sidewise slippage. A keel is a thin fin attached to the lower center line of the boat. Keels come in many shapes. They harness the viscosity of the water to oppose sidewise slippage. At the same time, they have very small resistance to the boats forward motion. That helps the boat to stay on course.
The rudder is another underwater, fin-like part, located at the stern of the boat. It can be turned right and left on a vertical axis. When the rudder is aligned with the keel, it acts as its continuation, and helps the boat keep moving on a straight line. When the rudder is turned from that alignment, the moving boat turns.
Figure 1 Keels and Rudders
The viscosity force reduces the sidewise slipping of a boat, but it cannot eliminate it completely. It also opposes the forward motion of the boat. In the following we will ignore sidewise slipping and water resistance to forward motion. These effects could be merged with our descriptions.
The wind that acts on the boat can blow in any direction, but the boat can move only in the direction of her keel. The situation is similar to a bead on a hard wire. The bead can move only in the direction of the wire, but the force on the bead may point to any direction. Consider a hard wire aligned in left-right direction, and an active force that acts on the bead (Figure 2, left). The bead would move to the left if the active force points straight to the left, or diagonally to the left. The effective force that drives the bead is only a part of the active force. This part of the force is called the component of the force in the direction of the wire. The acting force is most effective when it points in the direction of the wire. Its effectiveness decreases as the angle that it makes with the wire widens, and it becomes completely ineffective when it is perpendicular to the wire. The situation is symmetric for forces that point to the right.
The same ideas apply to the boat (Figure 2, right). The active force on the boat is the wind force. Only part of the wind force propels the boat in the direction of the keel. This part is the component of the wind force in the keel’s direction. We will call it here the heading force. The heading force is the force that actually propels the boat. The boat moves in the direction of the heading force. If the heading force points to the bow, the boat moves forwards. If that force points to the stern, the boat moves backwards. Like a bead on a wire, the strongest heading force occurs when to total wind force is aligned with the keel. As the angle between the total-wind-force and the keel widens, the heading force weakens. It vanishes when that angle between the keel and the total-wind-force reaches ninety degrees.
Drag and lift forces
It is common to see objects blown by the wind. The wind takes with it the leaves in the fall. It pushes them in the same direction that it is blowing. But besides pushing, winds exert on objects another type of force. The following simple experiment illustrates the other type of wind force.
Hold a piece of paper at the level of your eyes and blow into it. The air that flows from your mouth pushes the paper away from you. The force of this wind pushes the paper in the direction of the flow. This kind of pushing force is called in physics drag.
Now hold the same paper below your lips and blow (Figure 3). The air that comes from your mouth flows over the paper; it cannot push it. Still, the paper moves upwards, into the flowing stream of air. In this case, the flowing air sucks the paper. In physics, this kind of suction force is called lift.
Figure 3 lift force
Drag force is the component of the wind force in the direction of the wind. Lift force is the component of the wind force ninety degrees to the direction of the wind. The lift force that acts on the wing of airplanes lifts them and keeps them up, hence the term “lift”. However, lift forces can act in any direction with respect to the ground, depending on the object and the direction of the wind. Opposite lift forces are acting on the two sides of the sail. Their combined effect is to suck the front of the sail (Figure 4). In the following, we will refer to this sum as the lift force.
Wind exerts both drag forces and lift forces on objects. The total wind-force on the object is the sum of the drag and the lift forces. When lift force is present, the direction in which the wind is blowing is different from the direction of the total wind force that the wind exerts on an object. The direction of the total wind force depends on the relative strengths of the drag and the lift forces. When the lift force is relatively small compared to the drag, the total wind force is close to the wind’s direction. When the lift force is relatively large compared to the drag, the total force is almost perpendicular to the wind. When the sail is cutting straight into the wind it does not inflate, and lift is not created; only drag acts in such cases on the flapping, flat sail. The drag force is created mainly by wind blowing into the back of the sail. The lift force is created by wind that flows across the front surface of the sail. (Figure 5).
Figure 5 drag, lift, and total wind force in various attack angles
The direction of the total wind force is always between the directions of the drag and the lift forces. A sail on its own will always be pushed downwind; if it creates lift, it will be pushed diagonally downwind.
Sailing against the wind
A sailboat cannot sail straight into the wind. The drag force will push it downwind. In order to get from point A to a point B that is directly upwind, the boat must zigzag. It sails from A diagonally into the wind to a point C. At C it turns and sails diagonally into the wind to point B (Figure 6). The boat may zigzag several times on its way from A to B.
In order to sail from A to C, the boat is first turned towards C by using the rudder. Once the keel is aligned in the A to C direction, the rudder is aligned again with the keel, and the sail is set at an angle to the keel, creating a heading force that points to C. That drives the boat, like a bead on a wire, from A to C. Figure 7 shows the relationships between the directions of the wind, sail, keel (boat) and the forces as the boat sails diagonally into the wind. Once at C, using the rudder, the boat is turned towards B. The rudder is then aligned with the keel and the sail is allowed to swing to the other side of the boat, to a point where the heading force points to B. That drives the boat to B; again, like a bead on a wire. The entire configuration of boat-wind of figure 7 is turned around to the direction C to B; the only difference is the sail is now set on the other side of the boat’s center.
For a boat to sail diagonally into the wind, the sails must generate enough lift force, compared to the drag force that is always acting on them. The ratio between the strengths of the lift and drag forces depends on the wind and on the sail. In order to move into a steady wind (figure 7), the sail has to maintain the necessary attack angle (the angle that the sail makes with the wind) and its curvature. Otherwise, drag will be the dominant force. For example, a flag on the mast cannot serve as a sail, because it swings to the wind’s direction and it cannot maintain a curvature. It exerts on the mast a drag force in the wind’s direction. Triangular sails are “naturals” for creating significant lift force. It is simple to rig them (connect them to the boat) such that they maintain their curved shape and attack angle. Other sails can also create significant lift, as long as they maintain their curvature and attack angle. The U.S. Coast Guard Eagle in Figure 8 uses a variety of sails that cut into the wind, as she sails diagonally upwind.
Figure 8 USCG Eagle sailing diagonally into the wind
When sailing downwind, the wind blows into the back of the sails and inflates them. The sails “try to stop the wind”. This creates the drag force that pushes the boat. Some sails are designed for sailing both upwind and downwind, and other are specialized for up or downwind sailing.
Figure 9 sailing downwind
CAN SAILBOATS SAIL FASTER THAN THE WIND?
IN A NUTSHELL
A keel enables a sailboat to sail diagonally into the wind. As the sail moves into the wind, it “feels” a faster wind on its face. This is called the relative velocity of the wind with respect to the sail. A wind of larger relative velocity exerts a larger force on the sail, which accelerates the boat. The boat moves faster, which increases the relative speed of the wind, which increases the wind-force, which accelerates the boat, and on and on. However, the resistance of the water slows down the motion of the boat. Eventually, a balance is reached between the force of the wind and the force of the water, and the boat moves at a constant velocity, diagonally into the wind. That final constant velocity may be greater than the wind velocity with respect to the water. Whether a boat could reach such final velocity will depend on the characteristics of the boat.
The velocity of an object is defined as the distance that it covers divided by the time that it takes to cover that distance. If a boat covers 10 miles in an hour, her velocity is 10 miles per hour (mph). The relative velocity of two objects is defined as the change in the distance between them divided by the time it takes to create that change.
If two boats are moving in the same direction, their relative velocity is the difference between their individual velocities. For example, if both move at 10 mph in the same direction, the distance between them does not change, so, by definition, their relative velocity is zero. If we subtract their velocities we get the same answer: 10 mph-10 mph= 0 mph.
If the two boats are moving in opposite directions, their relative velocity is the sum of their individual velocities. For example, if one boat moves at 10 mph to the north and the other at 15 mph to the south, the distance that the first covers in an hour is 10 miles, the distance that the second covers is 15 miles, and the change in the distance between them is 10 miles + 15 miles =25 miles. This happens in one hour, so, by definition, their relative velocity is 25 mph. The first boat is moving with respect to the second at a relative velocity of 25 mph to the north. The second boat is moving with respect to the first at 25 mph to the south.
The largest relative velocity between two objects occurs when they move in exactly the opposite directions. The smallest relative velocity occurs when they move in exactly the same direction. When they move diagonally, their relative velocity is somewhere in between those minimum and maximum values; the exact value depends on the angle between their directions. When they move ninety degrees or more with respect to each other, their relative velocity is larger than each of their individual velocities. Figure 10 shows the relative velocity of the wind with respect to a moving sail, when the speed of the wind is 20 mph with respect to the sea, and the speed of the sail is 15 mph with respect to the sea.
Figure 10 wind’s velocity, sail’s velocity and wind’s velocity relative to the sail for different directions of motion of the sail. Length of arrow proportional to the velocity. Sail’s and boat’s velocities are the same.
The figure illustrates that the smallest relative velocity is when the sail and the wind are moving in the same direction, and the largest is when they move in opposite directions. The figure also shows that when the boat moves diagonally into the wind, the relative wind’s velocity is greater than the wind’s velocity with respect to the sea.
Relative wind-sail velocity
Both the drag and the lift forces depend on the relative velocity between the air and the affected object. It does not matter if the object is at rest, like a kite, and the air is moving; or the object is moving and the air is not moving, like an airplane in calm air; or the object and the air are moving, like a sail in the wind. Drag and lift forces vanish when the relative velocity is zero, and they get stronger as the relative velocity increases.
When the boat and the wind move in same direction, the relative velocity of the wind is the difference between the boat’s velocity and the wind velocity. Initially, as the boat moves slower than the wind, say at 1 mph in a wind of 6 mph, the relative wind velocity is 6-1=5 mph. The drag force accelerates the boat. When the boat reaches, say 4 mph, the relative velocity of the wind is only 6-4=2 mph. The drag force is now much smaller than at the beginning, but it still could accelerate the boat. If the boat reaches 6 mph, the relative wind velocity will be 0 mph, and the drag force will vanish. The air will not push the boat any more, and she will move at 6 mph - the wind velocity. In reality, the boat will move slower than the 6 mph of the wind. The wind’s force has to overcome the resistance of the water to the boat’s forward motion. The boat will move at constant velocity of less than 6 mph, as the forces of the wind and the water balancing each other.
The situation is different when the boat sails diagonally against the wind. The main force in such cases is the lift force created as the wind flows across the surface of the inflated sail. The sail and that wind move diagonally against each other. Therefore, the relative velocity of the wind is greater than the velocity of the boat, regardless of how fast the boat is moving. That will accelerate the boat, which in turn will further increase the relative velocity of the wind., and so on and on. Since the relative wind velocity is increasing as the boat moves faster, it will not reach zero and stop pushing the boat, as in the downwind case. However, the boat will eventually reach a maximum velocity. This will happen when the resisting force of the water becomes equal to the heading force created by the wind. The net force on the boat will then be zero, and she won’t move any faster. By then, she could be moving faster than the wind. One of the factors that affect the resisting force of the water is the submerged volume of the boat. The lighter the boat, the less water she has to push aside in order to move ahead. A light boat that sails diagonally into the wind, whose sails and underwater structures cut efficiently into the wind and the water, and whose sails provide enough heading force can sail faster than the wind (diagonally into the wind).
When sailing diagonally with the wind, a boat can also create relative wind-sail velocity that is greater than the relative wind-water velocity. In such cases, the lift force can become more significant than the drag force, similarly to sailing diagonally against the wind. However, the overall motion of the boat depends also on her sidewise slippage and the resistance of the water to her forward motion. In sailing with the wind, these two factors are more significant than in against-wind sailing. In general, boats move faster when they sail diagonally against the wind. Ice boats are like sail boats that slide on ice. They are fitted with skis or with blades that have very small resistance to forward motion and very large resistence to slippage. That enable them to sail diagonally with the wind many times faster than the wind.