1.0 Turning Effect of Force (Moment of Force)
In previous classes, we learned that a force can move an object or change its shape. In Class 8, we explore how a force can cause an object to rotate about a fixed point or axis. This turning effect is what allows us to open doors, use see-saws, or turn a steering wheel.
Moment of Force (Torque)
The turning effect of a force acting on a body about a fixed point (pivot or fulcrum) is called the Moment of Force.
- S.I. Unit: $N\,m$ (Newton-metre)
- C.G.S. Unit: $dyne\,cm$
- Dependence: It depends on the magnitude of the force applied and the perpendicular distance of the line of action of the force from the axis of rotation.
Formula for Moment of Force
$$\tau = F \times d$$
Where: $\tau$ = Moment of force, $F$ = Applied Force, $d$ = Perpendicular distance from pivot.
1.1 Direction of Rotation
The direction of the moment of force is classified based on the rotation it produces:
- Clockwise Moment: If the force tends to turn the body in the direction of a clock's hands. By convention, it is taken as Negative.
- Anti-clockwise Moment: If the force tends to turn the body opposite to a clock's hands. By convention, it is taken as Positive.
To produce a maximum turning effect with minimum force, you should apply the force at the maximum possible distance from the pivot. This is why door handles are always placed at the edge opposite to the hinges!
A force of 10 N is applied perpendicularly to a wrench at a distance of 25 cm from the nut. Calculate the moment of force produced.
Solution:
1. Given Force ($F$): $10\,N$
2. Distance ($d$): $25\,cm = 0.25\,m$ (Must convert to S.I. units!)
3. Formula: $\tau = F \times d$
4. Calculation: $\tau = 10 \times 0.25 = 2.5\,N\,m$.
Final Answer: The moment of force is $2.5\,N\,m$.
Archimedes once said, "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." He was referring to the incredible power of the Moment of Force!
2.0 Pressure: Thrust and its Effect
While Force can change the motion of an object, Pressure describes how that force is distributed over a surface. To understand pressure, we must first define the total force acting perpendicularly on a surface, known as Thrust.
What is Pressure?
Pressure is defined as the thrust acting per unit area of a surface.
- S.I. Unit: $Pascal$ ($Pa$) or $N/m^2$.
- 1 Pascal: It is the pressure exerted by a thrust of $1\,N$ on an area of $1\,m^2$.
- Dependence: Pressure is directly proportional to Thrust and inversely proportional to the Area of contact.
Pressure Formula
$$P = \frac{F}{A}$$
Where: $P$ = Pressure, $F$ = Thrust (Force), $A$ = Area ($m^2$).
2.1 Applications of Pressure
In our daily lives, we often manipulate the area of contact to either increase or decrease pressure:
- Increasing Pressure: Tools like knives, needles, and nails have extremely small areas at their tips so that even a small force creates enough pressure to pierce or cut.
- Decreasing Pressure: School bags have wide straps to distribute the weight over a larger area of the shoulder, reducing the pressure and pain. Similarly, heavy trucks have double wheels to reduce the pressure on the road.
When calculating pressure, ensure the area is in $m^2$. Students often forget to convert $cm^2$ to $m^2$.
Remember: **$1\,m^2 = 10,000\,cm^2$**. To convert $cm^2$ to $m^2$, divide by $10,000$.
A force of 50 N is applied on an object of area 2 cm². Calculate the pressure exerted.
Solution:
1. Force ($F$): $50\,N$.
2. Area ($A$): $2\,cm^2 = \frac{2}{10000}\,m^2 = 0.0002\,m^2$.
3. Formula: $P = F / A$.
4. Calculation: $P = 50 / 0.0002 = 2,50,000\,Pa$.
Final Answer: The pressure is $2.5 \times 10^5\,Pa$.
A flat-footed person exerts less pressure on the ground than someone wearing high-heeled "stiletto" shoes! This is because the tiny area of a heel concentrates the entire body weight into a very small point.
3.0 Liquid Pressure (Hydrostatic Pressure)
Unlike solids, which exert pressure only in the downward direction, liquids exert pressure in all directions. This is because liquid molecules are in constant motion and collide with the walls of the container and any object submerged within them.
Characteristics of Liquid Pressure
- Depth: Pressure increases with the depth from the free surface of the liquid.
- Density: Higher the density of the liquid, the greater is the pressure it exerts.
- Direction: At a given depth, the liquid exerts equal pressure in all directions (sideways, upwards, and downwards).
- Surface Level: A liquid seeks its own level, meaning pressure at all points on the same horizontal plane is equal.
Liquid Pressure Formula
$$P = h \times \rho \times g$$
Where: $h$ = depth (m), $\rho$ (rho) = density ($kg/m^3$), $g$ = acceleration due to gravity ($9.8\,m/s^2$).
3.1 Consequences of Liquid Pressure
The fact that pressure increases with depth leads to several important engineering considerations:
- Dams: The wall of a dam is made much thicker at the bottom than at the top to withstand the enormous pressure exerted by the water at great depths.
- Deep-Sea Diving: Divers must wear special pressurized suits. At extreme depths, the water pressure is high enough to crush a human body.
- Water Tanks: Storage tanks are placed at a height (like on a roof) so that the depth of the water column provides enough pressure for the water to flow out of the taps.
A common mistake is thinking that liquid pressure depends on the shape of the container or the total volume of liquid. It does not! Whether the tank is wide or narrow, the pressure at the bottom depends only on the vertical height ($h$) of the liquid column.
Calculate the pressure at the bottom of a swimming pool 5 m deep. (Density of water = $1000\,kg/m^3$, $g = 10\,m/s^2$)
Solution:
1. Depth ($h$): $5\,m$.
2. Density ($\rho$): $1000\,kg/m^3$.
3. Gravity ($g$): $10\,m/s^2$.
4. Formula: $P = h\rho g$.
5. Calculation: $P = 5 \times 1000 \times 10 = 50,000\,Pa$.
Final Answer: The liquid pressure is $50,000\,Pascal$ ($50\,kPa$).
Submarines have a "crush depth"—a specific depth beyond which the external liquid pressure becomes so great that the steel hull of the submarine will literally cave in!
4.0 Atmospheric Pressure
Just as water exerts pressure on a submerged object, the thick layer of air surrounding the Earth (the atmosphere) exerts pressure on everything on its surface. This is because air has mass and is pulled toward the Earth by gravity.
What is Atmospheric Pressure?
Atmospheric Pressure at any point is the force exerted per unit area by the weight of the air column above that point.
- Standard Value: At sea level, it is approximately $1.013 \times 10^5\,Pa$ (or 1 atmosphere).
- Variation with Altitude: It decreases as we go higher because the height of the air column above us decreases and the air becomes less dense.
Standard Atmospheric Pressure Units
$$1\,atm = 76\,cm\,of\,Hg = 760\,mm\,of\,Hg$$
Measured using a Barometer.
4.1 Simple Mercury Barometer
A barometer is an instrument used to measure atmospheric pressure. A simple mercury barometer consists of a long glass tube filled with mercury and inverted into a trough of mercury. The height of the mercury column supported by the atmosphere is a measure of the air pressure.
4.2 Common Applications
- Drinking Straw: When you suck through a straw, you reduce the air pressure inside. The higher atmospheric pressure outside pushes the liquid up the straw.
- Syringes: Pulling the piston creates a partial vacuum; atmospheric pressure then pushes the medicine into the syringe.
- Rubber Suckers: When pressed against a smooth surface, air is forced out. The external air pressure holds the sucker firmly in place.
You might wonder why $10^5\,Pa$ of pressure doesn't crush us. This is because our internal body pressure (blood pressure and air in our lungs/tissues) is equal to the external atmospheric pressure, balancing it out perfectly.
Why do mountaineers sometimes suffer from nosebleeds at high altitudes?
Solution:
1. At high altitudes, the atmospheric pressure is significantly lower than at sea level.
2. However, the blood pressure inside the human body remains higher (balanced for sea level).
3. This pressure difference causes the thin walls of the blood vessels in the nose to burst, leading to bleeding.
The Torricellian Vacuum is the empty space at the top of a mercury barometer. It is named after Evangelista Torricelli, who invented the barometer in 1643!