Pressure, Gravity and Moments Chapter 10

What is Density? The Density of a substance is its mass per unit volume . i.e.

What is the SI Unit of Density? The SI Unit of density is the kilogram per cubic metre ( kg m-3). Density is a Scalar Quantity..

SUBSTANCE DENSITY ( Kg m-3) Gold 19 320 Mercury (20 oC) 13 546 Lead 11 370 Steel 7 800 Glass 2 500 Wood (approx.) 550 Cork 250 Water (4 oC) 1 000 Sea Water 1 030 Carbon dioxide 1.977 Air (0 oC) 1.293

1 kg = 1 000 grams 1 gram = 1 × 10-3 kg 1 litre = 1 L = 1 000 cm3 1 m3 = 1 000 000 cm3 1 cm3 = 1 × 10-6 m3

What is Pressure? Pressure is force per unit area. i.e.

What is the SI Unit of Pressure? The SI Unit of pressure is the pascal (Pa). 1 pascal = 1 newton per square metre i.e. 1 Pa = 1 N m-2 Pressure is a Scalar Quantity.

Due to its weight the block exerts pressure on the surface underneath it. Force acting on surface = weight of block = (5)(9.8) = 49 N Area over which force acts = (0.04)(0.12) m2 = 0.0048 m2

Pressure in a Liquid The weight of the liquid is acting down on the base of the beaker. It exerts pressure on the base.

P =  g h At: Depth h in a liquid of: Density  Where: Acceleration due to Gravity is g, the Pressure P, due to the liquid is given by: P =  g h

Pressure in a liquid increases with depth. This follows from the formula P =  g h. It may be demonstrated as shown where the water from the deepest hole squirts out the farthest.

Pressure in a liquid acts perpendicular to any surface put in the liquid. This may be shown as in the diagram. The water emerges from each hole perpendicular to the surface.

Because pressure increases with depth, the pressure underneath the block is greater than the pressure on top. The liquid therefore exerts an overall upward force on the block.

Upthrust (Buoyancy Force) If an object is immersed in a liquid (or gas) there is an upward force acting on it due to the liquid (or gas). This force is called the Upthrust or the Buoyancy Force. Because pressure increases with depth, the pressure underneath the object is greater than the pressure on top. This causes the overall upward force on the object.

State Archimedes’ Principle Archimedes’ Principle states that when an object is immersed in a fluid it experiences an upthrust equal (in size) to the weight of the fluid it displaces.

The upthrust on the submerged part of the ship keeps it afloat

The Law of Flotation states that the weight of a floating body is equal to the weight of the fluid it displaces. The upthrust on the submerged part of the floating object is equal to the weight of the object.

As the Pressure of the trapped air is increased its Volume decreases.

State Boyle’s Law Boyle’s Law states that: for a fixed mass of gas at constant temperature, the Volume is inversely proportional to the Pressure. i.e. for a fixed mass of gas at constant temperature: or Where k is a constant

Boyle’s Law Apparatus

A graph of P against 1/V is a straight line through the origin. This shows that

Newton's Law of Universal Gravitation Newton's Law of universal gravitation states that any two point masses in the universe attract each other with a Force that is: Directly proportional to the product of their masses and Inversely proportional to the square of the distance between them.

and

Newton's Law of Universal Gravitation F is the force of attraction m1 and m2 are the masses d is the distance between the masses G is a constant called the Universal Gravitation Constant

The Gravitational Force is always attractive. The force on m1 pulls it towards m2 and the force on m2 pulls it towards m1. The Size of the Force on each mass is the same. This is true even if one mass is much larger than the other. The gravitational force between two spherical bodies is the same as if each had all its mass at its centre and the distance between the bodies is the distance between their centres.

The value of the Universal Gravitation Constant G is the same anywhere in the universe. The value of G is 6.7  10-11 N m2 kg-2. This is an extremely small number (0.000 000 000 067). Since G is such a very small number the gravitational force between two bodies is negligible unless at least one of them has a very large mass.

Newton’s Law of Universal Gravitation is an example of an Inverse Square Law. This means that the size of the force is inversely proportional to the square of the distance between the two bodies. i.e. Therefore: If the distance between the bodies is doubled the force becomes 4 times smaller. If the distance between the bodies is trebled the force becomes 9 times smaller, etc.

The force of gravity provides the centripetal force needed to keep the planets in orbit around the Sun. Note: Pluto has been reclassified as a dwarf planet.

The Moon’s Gravity is too weak to maintain an Atmosphere The Moon’s Gravity is too weak to maintain an Atmosphere. Any gas molecules placed on the Moon would escape into space.

It is much easier to open a door from a point on the door far away from the hinges than it is from a point near the hinges.

What is Moment of a force? The Moment (M) of a Force (F) about an axis is equal to the magnitude of the force multiplied by the perpendicular distance (d) from the axis to the line of action of the force.

Moment of force = Magnitude of Force  Perpendicular Distance M = F  d

The moment of a force is a Scalar Quantity. The SI Unit of moment of a force is the newton metre (N m).

Find the moment of a 10 N force about an axis that is 3 m from it. Moment of force = Force  Perp. Dist. from axis = 10  3 = 30 N m

Co-planar Forces Forces acting on a body are said to be Co-planar if they all act in the same plane.

Moments about A Moments about B The 20 N, the 30 N and the 60 N forces have a Clockwise Moments about A. The 50 N and the 80 N have an Anticlockwise Moments about A. Moments about B The 60 N and the 50 N forces have a Clockwise Moments about B. The 20 N, the 30 N and the 80 N have an Anticlockwise Moments about B

Equilibrium What is meant by saying that a Body is in Equilibrium? A body is said to be in Equilibrium if the body as a whole is not accelerating and the body is not rotating with angular acceleration. What is the difference between Static Equilibrium and Dynamic Equilibrium? If a body is not moving in any way it is in Static Equilibrium. If a body is moving in a straight line at a steady speed and is not rotating or is rotating with constant angular velocity, it is said to be in Dynamic Equilibrium.

State the two conditions necessary for a body to be in Equilibrium. What are the two conditions necessary for the Equilibrium of a set of co-planar forces? Or State the two conditions necessary for a body to be in Equilibrium. A body will be in Equilibrium if: The vector sum of the forces in any direction is zero. (ii) The sum of the moments about any point is zero.

A metre stick is in equilibrium under the forces shown A metre stick is in equilibrium under the forces shown. Find the value of the force X. Anticlockwise moments about 50 cm mark = (3030) + (2010) = 1 100 N m Clockwise moments about 50 cm mark = (X)(30) Since the metre stick is in equilibrium these are equal, thus: 30 X = 1 100 X = 36.67 N

The metre stick is in equilibrium. Find its weight.

To Investigate the Laws of Equilibrium for a Set of Coplanar Forces

If the man is just able to move the boulder with the crow-bar, what force does he exert?

What is a Couple? Two Parallel Forces with the same Magnitude acting in opposite directions is called a Couple.

The resultant of the two forces of a Couple is zero. A Couple therefore has only a turning effect. Since a couple has only a turning effect it has a turning moment. The moment of a couple about any point in its plane is the same. The Moment of a Couple (T) is sometimes called the Torque of a Couple.

Moment of couple (Torque) = T = F d The Moment of a Couple is the product of one of the forces and the perpendicular distance between them, i.e. Moment of couple (Torque) = T = F d

Moment of Couple (Torque) T = F d = (400)(0.4) = 100 N m A wrench is used to tighten a wheel nut on a car wheel. Two forces of magnitude 400 N acting in opposite directions are applied to the wrench as shown. Calculate the moment of the couple (i.e. the torque) they exert on the nut. Moment of Couple (Torque) T = F d = (400)(0.4) = 100 N m

Find the Moment of the Couple acting on the stick shown. Moment of Couple (Torque) T = F d = (400)(0.8) = 320 N m