• Describe the laws of motion, kinematics, dynamics and energy of the system

• Define upthrust in fluids, Archimedes principle, and floatation

• Understand physics principles and apply the conservation laws 

• Solve problems in fields of General Physics by using laws of physics

Kinematics is the study of motion without regard to the forces affecting it.

3. There are two types of quantities : (a) basic quantities and (b) derived quantities.

There are two kinds of physical quantities. They are scalar quantity and vector quantity.

(a) The study of ____ and ____ is physics.

(b) A ____ quantity is a physical quantity.

(c) There are two types of quantities, ____ quantities and ____ quantities.

(d) If a quantity has only magnitude, it is ____ and if it has both ____ and ____ it is a vector.

Column 1                  Column 2                                   

(i)    1 m                (a) 10-2 m 

(ii)   1 cm               (b) 103 g

(iii)  10-3 kg                   (c)  1 g

(iv)  1 kg               (d)  102 cm

(v)   1 kg m s-1            (e)  105 g cm s-1

Circular motion always involves a change in the direction of the velocity vector, but it is also possible for the magnitude of the velocity to change at the same time. Circular motion is referred to as uniform if |v| is constant, and nonuniform if it is changing.

Uniform circular motion can be described as the motion of an object in a circle at a constant speed. As an object moves in a circle, it is constantly changing its direction. At all instances, the object is moving tangent to the circle. Since the direction of the velocity vector is the same as the direction of the object's motion, the velocity vector is directed tangent to the circle as well.

An object moving in a circle is accelerating. Accelerating objects are objects which are changing their velocity, either the speed or the direction. An object undergoing uniform circular motion is moving with a constant speed. Nonetheless, it is accelerating due to its change in direction. The direction of the acceleration is inwards.

The final motion characteristic for an object undergoing uniform circular motion is the net force. The net force acting upon such an object is directed towards the center of the circle. The net force is said to be an inward or centripetal force. Without such an inward force, an object would continue in a straight line, never deviating from its direction. Yet, with the inward net force directed perpendicular to the velocity vector, the object is always changing its direction and undergoing an inward acceleration.

For nonuniform circular motion, the radial component of the acceleration ar obeys the same equation as for uniform circular motion. Since the velocity also changes in magnitude, the acceleration vector also has a tangential component at. The total acceleration vector a can be written as the vector sum of the component vectors.

An object in nonuniform circular motion has a changing speed and a changing angular velocity.

The tangential component of velocity is    vt = rω

The tangential acceleration is the rate of change of the tangential velocity, so

                                             at  = rα

The mathematical relationships between θ, ω and α are the same as the mathematical relationship between s, v and a.

Relationship between θ, ω and α for constant angular acceleration:

ω = ω0 + αt

                               q = ω0t +  (1/2) αt2

ω2 = ω02 + 2α (θ – θ0)

v =  1/2(ω0 + ω )

A disk with a 1.0 m radius reaches a maximum angular speed of 18 rads-1 before it stops 35 revolutions (220 rad) after attaining the maximum speed. How long did it take the disk to stop?

An object of mass 2 kg is moving in a circle of radius 2 m. At a given instant the object has a speed of 3 m/s and an angular acceleration of 2 rad/s2. Find the magnitude of the net force on the object at this instant.

The faster a ball is thrown upwards, the higher it rises before it is stopped and pulled back by gravity.

We will show that to escape from the earth into outer space an object must have a speed just over 11kms-1 (11,000ms-1) – called the escape speed.

The escape speed is obtained form the fact that the potential energy gained by the body equals its loss of kinetic energy, if air resistance is neglected.

The work done measures the energy transferred

Let m be the mass of the escaping body and M the mass of the earth, The force F exerted on the object by the earth is

The centripetal force which keeps an artificial satellite in orbit around the earth is the gravitational attraction of the earth for it.

For a satellite of mass m travelling with speed v in a circuit ar orbit of radius R (measured form the centre of the earth)

• Artificial satellites circling the Earth are now commonplace.

• A satellite is put into orbit by accelerating it to a sufficiently high tangential speed with use of rockets

• If the speed is too high the spacecraft will not be confined by the Earth's gravity and will escape never to return.

• If the speed is too Low, it will return to Earth.

• Satellites are usually put into circular (or nearly circular) orbits, because they required the Least take off speed.

• If is sometimes asked:" What keeps a satellite up?"

• The answers is its high speed. If a satellite stopped moving, it would, of course fall directly to Earth.

But at a very high speed a satellite has, it would quickly fly out into space (see figure – 2) if it weren't for the gravitational force of the earth pulling it into orbit.

In fact, a satellite is falling (accelerating toward Earth). But its height tangential speed keeps it from hitting Earth.

A satellite is any object that is orbiting the earth, sun or other massive body. Satellites can be categorized as natural satellites or man-made satellites. However, every satellite's motion is governed by the same physics principles. The fundamental principle to be understood concerning satellites is that a satellite is a projectile. That is to say, a satellite is an object upon which the only force is gravity. Once launched into orbit, the only force governing the motion of a satellite is the force of gravity.

The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The velocity of the satellite would be directed tangent to the circle at every point along its path. The acceleration of the satellite would be directed towards the central body that it is orbiting. And this acceleration is caused by a net force that is directed inwards in the same direction as the acceleration.

Rotational motion is more complicated than linear motion, and only the motion of rigid bodies will be considered here. A rigid body is an object with a mass that holds a rigid shape. Many of the equations for the mechanics of rotating objects are similar to the motion equations for linear motion.

Moment of inertia is a measure of an object's resistance to rotation. It is the rotational analog of mass for linear motion. It appears in the relationships for the dynamics of rotational motion. The moment of inertia must be specified with respect to a chosen axis of rotation. For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis,

                                             I = mr2                 

That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point mass. Therefore, the moment of inertia about an axis of a number of particles of mass is the sum of the moments of inertia of the individual particles.

                                        I = Ʃmiri2

Three small bodies, which can be considered as particles, are connected by light rigid rods as in figure. What is the moment of inertia of the system (a) about axis through point A, perpendicular to the plane of the diagram, and (b) about an axis coinciding with the rod BC?

The parallel axis theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes.

                                                         I = Icm + mr2

where, Icm is the moment of inertia of the object about an axis passing through its centre of mass, m is the object's mass and r is the perpendicular distance between the two axes.

Angular momentum is the momentum that a body has due to its rotation about an axis. It is a conserved vector quantity. A particle of mass m and velocity v has linear momentum p = mv. The particle may also have angular momentum L with respect to a given point in space. If r is the vector from the point to the particle, then 

The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object. The kinetic energy of a rotating object is analogous to linear kinetic energy and can be expressed in terms of the moment of inertia and angular velocity. The total kinetic energy of an extended object can be expressed as the sum of the translational kinetic energy of the center of mass and the rotational kinetic energy about the center of mass. For a given fixed axis of rotation, the rotational kinetic energy can be expressed in the form

KErotation =1/2 I ω2

In the absence of outside forces angular momentum is conserved, or it does not change. So a spinning object when free from any outside forces will continue to spin, it will conserve its momentum. One interesting byproduct of angular momentum and one of extraordinary importance is the gyroscope.

Due in great part to the conservation of angular momentum a gyroscope will maintain is relative position, and is very resistant to external torque. The reason for this is if one visualizes the mass of the gyroscope moving along a consistent circular path around the center of the gyroscope, in order to have this mass turn, or curve away from the plane of spin a force is required to accelerate the mass in a new direction. This is akin to how the law of inertia stated that a body in motion will tend to stay in motion. So, the inertia of the spinning gyroscope will tend to move into is circular path and resist movement or forces away from its plane of motion.

When a force applied to a gyroscope axis, it will move in a direction at right angles both to the plane of the spinning wheel and the plane in which the force is applied. This is called precession. The direction in which it will swing depends on the direction the rotor is turning. This motion is the result of the force produced by the angular momentum of the rotating body and the applied force.

The motion of a body with the uniform velocity (or) the constant velocity is called the uniform motion.      

Note:  For a uniform motion there is no acceleration ( a = 0).

Example: The displacement of a body for every second is shown below.

If a body does not have the same displacement in the same interval, its motion is not the uniform motion but it is an accelerated motion. (OR)

If the velocity of a body changes, its motion is called the accelerated motion.

Example: The displacement of a body for every second is shown below.

If a body moves along a straight line with the constant acceleration, the average velocity is given by

If a body moves along a straight line, its motion is called the linear motion (or) the straight line motion.

Note: For the linear motion the speed is the magnitude of velocity.

(For linear motion with uniform or constant acceleration 

If the change in velocity is constant for equal intervals of time, it is called uniformly accelerated motion. The acceleration is known as uniform acceleration

Consider a steel ball which is moving along a straight path with constant speed. It is a uniform motion. Let the speed of the ball be 5 cms-1. Then, the ball travels a distance 5 cm in every second. The distances and the times elapsed are given in the Table A.

(1) The distance-time graph is a straight line for uniform motion (no acceleration).

(2) The slope (slant) of the straight line gives the speed of the object. ( Speed = slope = tan )

The steeper the slope, the greater is the speed.

We consider a body moving with a constant (uniform) velocity of 5 m s-1 along a straight line. The speed is not changing, it is 5 m s-1 for every second. The body has no acceleration (a = 0). The speed-time graph is shown in Fig.1. The speed-time graph is a straight line  which is parallel to the horizontal axis (the time axis).

We consider a car travelling along a straight road with uniform acceleration. The  speedometer  shows  the  speed  of the car at the particular instant the reading is taken. They are given in Table A.

The  speed-time  graph  for  a  body  moving along a straight line with non-uniform acceleration (changing acceleration) is shown in the figure.

(1) The speed-time graph is a straight line for uniform motion and uniform accelerated motion.

(2) The slope of the speed-time graph gives the acceleration of the body.

(3) The area under the speed-time graph between two instants of time gives the distance.

(4) The speed-time graph is a curve for motion with non-uniform acceleration.

1. An object moves with an initial velocity of  5 m s-1. After 10 s its velocity is10 m s-1. If the object moves with constant acceleration in a straight line, find (a) its average velocity, (b)  the distance traveled in 10 s and  (c) its acceleration.

2. A particle with initial velocity of 10 m s-1 travels in a straight line and stops completely after 12 s. Find the uniform acceleration of the particle.

3. A particle starting from rest moves along a straight line with a constant acceleration of  2 m s-2. What is the velocity of the particle  9 s  after it started from rest?

4. A car is travelling with a constant velocity of 6 m s-1. The driver applies the brakes as he sees a cow which is at a distance of 24 m from the car.  Find the acceleration of the car if it stops just in front of the cow.

5. A body moving with a constant acceleration reaches the velocity of 4 m s-1 after 10 s. If the body starts from rest, find the respective velocities after (a) 12 s (b)  14 s (c)  16 s and (d)  18s. Draw  the  velocity-time graph for the                     interval10 s to 18 s . From  the graph, find the velocities at 8 s and 20 s  respectively.

6. Draw a graph of velocity against time for a body which starts with an initial velocity of 4 ms-1 and continues to move with an acceleration of 1.5 m s-2 for 6 s. Show how you would find from the graph: (a) the average velocity, and (b) the distance moved in those 6 s.

7. A car accelerates from  4 m s-1 to 12 m s-1 in 8 s.  Which graph shows the speed of the car varies with time?

The motion of a body round a circular path is the circular motion.  

Examples: 

(1) The revolution of the earth around the sun-  

(2) The rotations of the second hand, the minute hand and the hour hand.

If a particle rotates in a circle of radius r and moves along an arc length s, the angular displacement θ is given by

The angular velocity is the ratio of angular displacement to the time taken.

Parametric rolling is a dangerous and sudden increase in rolling motion caused by wave-induced variations in stability. This phenomenon typically occurs in head seas (waves coming from the bow) or following seas (waves from the stern).

✅ Wave Action on the Hull: Large waves cause the ship’s stability to change periodically.
✅ Asymmetry of the Hull: When a ship pitches in high seas, the waterline length changes, affecting buoyancy.
✅ Resonance Effect: Rolling amplifies when the natural roll period of the ship matches the wave encounter period.

⚠️ Extreme Rolling Angles (exceeding 30°) – Risk of capsizing.
⚠️ Cargo Shifting & Container Loss – Dangerous for container ships and bulk carriers.
⚠️ Structural Damage – Increased stress on hull and deck structures.
⚠️ Crew Injuries – Sudden violent rolling makes working conditions hazardous.

✅ Adjust Speed & Course – Avoid wave encounter periods that match the ship’s roll period.
✅ Use Ballast Adjustments – Change the ship’s stability characteristics to alter roll frequency.
✅ Active Stabilization Systems – Use stabilizers or anti-roll tanks.
✅ Weather Routing & Avoidance – Avoid known conditions that induce parametric rolling.

Synchronous rolling occurs when the wave encounter period coincides with the natural rolling period of the ship, causing the vessel to roll continuously and dangerously. Unlike parametric rolling, synchronous rolling happens in beam seas (waves striking from the side).

✅ Beam Seas: Waves coming directly from the side induce rolling.
✅ Resonance Effect: The ship’s roll motion synchronizes with the wave period.
✅ Lack of Course Correction: If the ship maintains a fixed course, rolling effects worsen.

⚠️ Severe Rolling (> 40°) – Risk of capsizing in extreme cases.
⚠️ Loss of Stability – Increased risk of losing control.
⚠️ Cargo Damage & Shifting – Can cause container collapse and dangerous deck conditions.
⚠️ Crew Fatigue & Seasickness – Makes operations challenging.

✅ Change Course – Alter heading to reduce beam sea exposure.
✅ Adjust Speed – Reduce wave impact by adjusting speed.
✅ Use Anti-Roll Devices – Deploy stabilizers or bilge keels.
✅ Secure Cargo Properly – Ensure lashings and securing systems are optimal.

(1) The disk or wheel rotates five revolutions each second, it has an angular velocity of

(a) 5 rps    (b) 2.5 rad s-1    (c) 10 deg s-1    (d) 2 rpm.

(2) If a flywheel whose diameter is 0.2m is rotating at 120 rpm, the linear velocity of a point on the rim is

(a) 0.4  m s-1    (b) 4  m s-1      (c) 0. 8  m s-1  (d) 8  m s-1

(3) A pendulum of length  0.5 m  travels along a path from A to B as shown in figure. The angle covered by the cord is 

(a) 0.4 rad   (b) 1.0 deg   (c) 0.4 rev    (d) 2 rev.

Dynamics is the study of forces and motion. (or) Dynamic is the study of the effect that forces have on the motion of objects.

Mass of an object is a measure of the quantity of matter in it. Mass is a basic property of the body. It does not change according to location, shape and speed (for speeds much less than the speed of light).

The  momentum of a moving object was defined as the product of its mass and its velocity.

If there is no net external force acting on a system consisting of two bodies, the sum of the momentum of the two bodies will remain constant. 

The total momentum  before collision of the two bodies is equal to their total momentum after collision.

Elastic collisions are those in which kinetic energy is conserved.

By principle of conservation of momentum:

Inelastic  collisions  are  those  in which kinetic energy is not conserved. It may be converted to internal energy, elastic potential energy of deformation, heat and sound energy.

By principle of conservation of momentum:

m1u1 + m2u2 = m1v1 + m2v2 

When  the  two  bodies  stick together after impact, they move off with a common velocity. This form of inelastic collision is called completely or perfectly inelastic collision.

In this case, conservation of momentum gives:

m1u1 + m2u2 =(m1 + m2) v

A 6 kg block moving with a velocity of 5 ms-1 collides with a 8 kg block moving in the opposite direction at 3 ms-1. If 6 kg block moves on with a velocity of 0.5 ms-1 after collision, calculate the velocity of 8 kg block.

By principle of conservation of momentum,

         Momentum before collision = Momentum after collision

                       m1u1 + m2u2 = m1v1 + m2v2

                       (6 × 5) + (8× (-3))  = (6×0.5)+ 8v2

                                   6      =  3+ 8v2

                                   v2 = 0.38ms-s

The 8 kg block moves on with a velocity of 0.38 ms-1 after collision.

A particle of mass m moving with speed u makes a head-on collision with an identical particle which is initially at rest. The particles stick together and move off with a common velocity. Find the final speed of the particles after the collision.

By principle of conservation of momentum,

                                 m1u1 + m2u2 =(m1 + m2) v

Since m1 = m2 = m and one of the particles is initially at rest,

                                 mu + 0 = 2mv

v = u/2

1. True or False?

(a) Mass is a vector quantity.

(b) Momentum is a scalar quantity.

(c) The unit of momentum in SI system is kilogram per square metre.

(d) Momentum of an object at rest is zero.

2. A light truck of 4000 kg moving with 10 m s-1 (E) slows down to 5 m s-1 (E). Its change in momentum is 

(a) -20,000 kg m s-1 (b) 20,000 kg m s-1 (c) 10000 kg m s-1 (d) -10,000 kg ms-1.

3. A ball 'A' of mass 100 g moving with a velocity of 5 m s-1 makes a head on collision with a ball B of mass 200 g moving with a velocity of 1 m s-1 in the opposite direction. If A and B stick together after collision, compute their common velocity v in the direction of A.

4. A 10 g bullet is shoot out of a pistol with speed 100 m s-1. Calculate the recoil velocity of  the pistol of mass 1kg.

If  there  is  no  net  external  force act upon it, a particle at rest will remain at rest and a particle in motion with a uniform velocity will continue to move with same constant velocity. 

In mathematical form,

The acceleration of a body is directly proportional to the net force applied on it and inversely proportional to the mass of the body.

A pendulum is a mass (bob) suspended from a fixed point that swings back and forth under the influence of gravity. The motion of a pendulum follows simple harmonic motion (SHM) when small oscillations occur.

Types of Pendulums

1. Simple Pendulum – A mass suspended by a string or rod that swings in a single plane (e.g., a ship’s inclinometer).

2. Compound Pendulum – A rigid body swinging about a pivot point (e.g., the movement of a ship’s mast in waves).

3. Foucault Pendulum – A long pendulum used to demonstrate the Earth’s rotation.

Ship Stability and the Metacentric Height (GM)

A ship behaves like a large pendulum in rolling motion. The stability of a ship is related to its metacentric height (GM):

• Large GM → Faster rolling motion (higher frequency, ship feels stiff).

• Small GM → Slower rolling motion (lower frequency, ship feels tender).

• Negative GM → Ship is unstable and may capsize.

✅ Marine Chronometers: Early ship clocks used pendulums for timekeeping, but they were later replaced by quartz and atomic clocks due to ship movement interference.
✅ Inclinometers (Heel Indicators): Pendulums help measure the tilt (heel) of a ship in waves, assisting in stability monitoring.
✅ Foucault Pendulum and Gyroscopes: Used in navigation to demonstrate the Earth’s rotation, influencing gyrocompass function.

 Practical Considerations for Deck Officers

• Observing ship rolling behavior helps in assessing stability.

• Using inclinometers and pendulum-based devices improves navigation accuracy.

• Applying pendulum motion principles ensures cargo stability and safe maneuvering.

A constant force acts on a 5. 0 kg object and reduces its velocity from 7.0 m/s to 3.0 m/s in a time of 3.0 s. Find the force.

We must first the acceleration of the object, which is constant because the force is constant. Taking the direction of motion as positive,

A book sits on a horizontal top of a car as the car accelerates horizontally from rest. If the static coefficient of friction between car top and book is 0.45, what is the maximum acceleration the car can have if the book is not to slip?

For the pulley system, find the acceleration, a and the tension, T in term s of masses m1, m2 and the acceleration due to gravity, g.

Whenever two particles interact, the force exerted by the second on the first is  equal  in magnitude and opposite in direction to the force exerted by the first on the second.

All  bodies attract one another with a force which is proportional to the product of their masses and inversely proportional to the  square of the distance between them.

In mathematical form,

Mass is the quantity of matter in a body. Mass is also a measure of inertia. It does  not change according to location, shape and  speed (for speeds much less than the speed of light). 

The weight of a body is defined as the attracting force of the earth acting on the body.

W = mg

W = Weight; m = mass of the body; g = acceleration due to gravity

When a body of mass  m falls freely on the earth’s surface, the gravitational field exerts a force on the particle. The gravitational field g, also known as the acceleration due to gravity is defined as:

Compute the mass of the Earth, assuming it to be a sphere of radius 6370 km. Give your answer to three significant figures.

An object weighs 100 N on the surface of the earth. What is the gravitational force that the earth exerts on this object when the object is at a position that is two earth radii from the center of the earth?

The acceleration due to gravity g is measured from a set of axes whose origin is at the  center of the  earth. But this does  not  rotate with the earth. The value of g measured by these "fixed" axes may be termed the absolute value of g. Because the earth rotates, the acceleration of a freely falling body as measured from a position on the surface of the earth is slightly less than the absolute value.

The force of gravitational attraction of the earth on a body depends on the position of the body relative to the earth. If the earth were a perfect homogeneous sphere, a body with a mass of exactly 1 kg would be attracted to the earth by a force of 9.825 N on the surface of the earth. But this value changes with respect to altitude.

While altitude is often associated with high elevations like mountains and aircraft, it also plays a role in maritime physics, particularly in ship performance, atmospheric pressure, and oceanographic conditions.

At sea level, atmospheric pressure is highest because there is the maximum amount of air above. Standard atmospheric pressure is:

P0=101.325kPa (or 1 atm)

🔹 Ships and maritime weather systems depend on atmospheric pressure changes for navigation, weather forecasting, and stability.

• High-Pressure Zones (Anticyclones) → Clear skies, calm seas.

• Low-Pressure Zones (Cyclones) → Storms, rough waves, and potential hurricanes.

• Barometric Pressure Readings help predict changing weather conditions for ship safety.

The ocean absorbs and retains heat, influencing global climate patterns. However, temperature changes at different sea altitudes (depths and high waves) affect maritime conditions.

Temperature Variation by Altitude and Depth

• Near the surface: Warmer due to direct sunlight.

• At depth: Cooler due to limited sunlight penetration.

• Higher waves (altitude in storms): Cooler due to increased exposure to wind.

🔹 Higher waves in storms create colder sea spray, affecting ship deck temperatures.

Gravitational acceleration varies slightly with altitude and latitude. The Earth's shape (an oblate spheroid) means that gravity is stronger at the poles and weaker at the equator.

g=9.81 m/s2 at sea level

🔹 In maritime navigation, this variation is minimal but important for precision navigation systems (like GPS and ship stabilization).

• Higher Altitudes (High Waves, Storms) → Instability

  • Large waves cause vertical movement, affecting cargo stability.

  • Shipboard instruments (barometers, GPS) must adjust for altitude effects.

• Lower Altitudes (Deeper Waters) → Increased Pressure

  • Submarines experience high hydrostatic pressure at deep-sea altitudes.

  • Pressure increases by 1 atm every 10m below sea level.

At sea level (0m altitude), water boils at 100°C under normal pressure.
However, in low-pressure storm systems, a slight decrease in boiling point may occur.

If  you  press against a wall with one of your hand, an opposite force of the same magnitude acts from the wall on the hand. If you press harder, the wall does the same. When two people pull on two spring scale, regardless of who tries to pull harder, the two forces as measured on the spring scales are equal.

The normal force acts on any object that touches surface. It always acts perpendicularly to the surface. The formula to calculate the normal force is

N = -mg

 where, N is normal force in Newton (N),

m is the mass in kg, and

g is the gravitational force in m/s2.

The frictional force (f) is proportional to the normal force, this means that f α N

Friction is a force which opposes motion. 

The direction of friction is opposite with the direction of motion.

Static friction: resists an object to start moving or sliding

Kinetic friction: resists an object that is already moving or sliding and always acts in a direction opposite of the motion

The magnitude of the force of static friction between any two surfaces in contact can have the values

Fs ≤ μs N 

where,  the  dimensionless  constant  µs   is  called  the  coefficient of  static friction  and N is the magnitude of the normal force exerted by one surface on the other. The magnitude of the force of kinetic friction acting between two surfaces is

Fk ≤ μk N 

where, µk is the coefficient of kinetic friction.

Note: 

• Static friction is always stronger than kinetic friction

• Friction always opposes the motion or attempted motion of one surface across another surface.

• Friction is dependent on the texture of both surfaces.

• Friction is also dependent on the amount of contact force pushing the two surfaces together

• It is independent of the area of the surfaces in contact

A 32 N block placed on a horizontal surface is pulled by a rope horizontally. The tension in the rope can be increased to 8 N before the block starts to move. A force of 4 N will keep the block moving at constant speed once it has been set in motion. Determine the coefficients of static and kinetic friction.

                                 Fs = μs N

                                 8 = μs (32)

The coefficient of static friction  μs = 0.25      

                                 Fk = μk N

                                 4 = μk (32)

The coefficient of kinetic friction  μk 0.125

When contact forces are applied to an object, the object is deformed. But the object returns to its original shape and size when applied forces are removed. Many solid are so rigid that deformation cannot be seen with human eye.

Elasticity: Elasticity is the ability of a substance to recover its original shape and size after distortion.

Stress: Stress is the deforming force acting on an object per unit cross-sectional area. Stress produces a strain.

Strain: Strain is a measure of the degree of deformation.

The lengths of the same spring stretched by a load of mass 40 g and 55 g are shown below. What is the length l if the spring constant, k is 4.9 Nm-1?

(1) Action and reaction (a) acts upon two different bodies (b) can be canceled out each other  (c) acts upon a single body (d) acts one after another. [Choose the correct answer]

(2) A wooden block is suspended as shown. The force T is the tension of the cord, its reaction is  (a) the weight W (b) tension acted upon the cord by the block (c) the gravitational force of attraction by the block on the earth (d) none of (a), (b) and (c).

Consider a man, pulling a wagon with a constant force F. If the man pulls the rope with a force F of 20 N at an angle of 30˚ to the ground, find the work done by the man to move the wagon through a distance of 10 m.

Resolving the force F in the direction of displacement s

W =   (F cos θ) x s

=  20 cos 3 0˚ x 10     

 =  173J

A forklift lifts a crate of mass 100 kg at a constant velocity to a height of 8 m over a time of 4 s. The forklift then holds the crate in place for 20 s. (a) Calculate how much power the forklift exerts in lifting the crate? (b) How much power does the forklift exert in holding the crate in place?

A stone of mass m has 100 J of potential energy when it is at a height h above the ground. What will be its potential energy if its height above the ground is halved?

Comet Shoemaker-Levy, which struck the planet Jupiter in 1994, had a mass of roughly 4x1013 kg, and was moving at a speed of 60 km/s. Compare the kinetic energy released in the impact to the total energy in the world’s nuclear arsenals, which is 2x1019 J. Assume for the sake of simplicity that Jupiter was at rest.

The  energy in the universe is constant. Energy cannot be made or destroyed, only changed  from one form to another form. This law applies to any closed system. A closed  system is a system  where no energy leaves the system and goes into the outside world, and no energy from the outside world enters the system. Therefore, the sum of the kinetic energy and the potential energy in a closed system is constant. It is the law of conservation of mechanical energy.

A student drops an object of mass 10 kg from a height of 5 m. What is the velocity of the object when it hits the ground? Assume, for the purpose of this question, that g = –10 m/s2.

When the object is dropped, it has a gravitational potential energy of:

                                             PE = mgh= (10kg) (-10ms-2) (-5m)

                                               = 500J

By the time it hits the ground, all this potential energy will have been converted to kinetic energy.

Any  machine which does work uses energy. But not all machine used by the machine is used as work, some will be wasted. The wasted energy usually results from frictional forces in the machine which produce heat energy.

The efficiency of a machine can be given by:

A hoisting machine lifts a 3000 kg load a height of 8.00 m in a time of 20.0 s. The power supplied to the engine is 18.0 hp. Compute (a) the work output (b) the power output and power input, and (c) the efficiency of the engine and hoist system.

1) A body is in equilibrium condition if there is no net force applied on it.

If several forces are exerted on a particle, they can be compounded by vector methods. The sum of all the forces is called the resultant force, or resultant. When two forces F1 and F2 are concurrent they can be added together to form a resultant FR = F1 + F2 using the parallelogram law.

The screw eye in figure is subjected to two forces, F1 and F2. Determine the magnitude and direction of the resultant force.

The  moment of a force, also known as the torque τ is defined as the force F acting at a perpendicular distance d.

  τ =Fd 

τ   is a vector and the SI unit for torque is the newton metre (N·m).

Consider a particle of mass m moving in a circle of radius r. Suppose the tangential force F acts on the particle and accelerates it. Then,

F=ma (or) Fr = mra

Fr =mr2α

Since τ=Fd =Fr and I = mr2,

τ = l α

A special case of  moments is a couple. A couple consists of two parallel forces that are equal in  magnitude, opposite  in sense and do not share a line of action (see in figure 1.12).

It does not produce any translation, only rotation. The resultant force of a couple is zero. But, the resultant of a couple is not zero.

Two oppositely directed force s of equal magnitudes are applied perpendicularly to the ends of a rod of length L. The rod is pivoted a distance x from one end. What is the torque about the pivot P? Does the result depend on the value of x.

Center of gravity is a point at which all of the weight of an object appears to be concentrated. If an  object  rotates  when  thrown, the center of gravity is also the center of rotation. When an object is suspended so that it can move freely, its center of gravity is always directly below the point of suspension. In a uniform gravitational field, the center of m ass serves as the center of gravity.

Center of gravity of a lamina can be found easily by using plumb line. A lamina is hung  a   pin so that it can swing freely and finally come to rest when the center of gravity  is  vertically below the pin. A plumb line is hung in front of the lamina so that  we  can  draw  a vertical line downwards through the pin h ole. The center of gravity is somewhere along this line. Then the lamina is hung from a different point and  another line is obtained by doing above procedure again. Center of gravity of the lamina lies on a point where the two lines a cross.

Stability is a measure of a body’s ability to maintain in its original position. A pencil balanced upright on its tip could theoretically be in equilibrium, but even if it w as initially perfectly balanced, it would topple in response to the first air current or vibration from a passing truck. The pencil can be put in equilibrium, but not in stable equilibrium.

What is a Point Load?

A point load is a force applied at a single, specific location on a structure or surface. In ships, point loads occur when:
✅ Heavy cargo is placed at a single location on the deck.
✅ A crane applies force on a single lifting point.
✅ Mooring lines exert tension at specific points.
✅ A wave impact concentrates force at one part of the hull.

🔺 Local Structural Stress – Can cause buckling or indentation if the structure isn't reinforced.
🔺 Uneven Weight Distribution – May create excessive hogging or sagging.
🔺 Instability – Affects ship balance, causing list or trim.

Deck Officers must ensure that point loads are well distributed across the ship’s structure to avoid excessive stress.

Why is Weight Distribution Important?

The way weight is spread across a vessel affects stability, fuel efficiency, and safety. Proper weight distribution ensures:
✅ Stable trim and list (prevents excessive tilting).
✅ Even stress on the hull (avoids structural fatigue).
✅ Efficient fuel consumption (reduces resistance and drag).

1. Center of Gravity (COG) – The point where the entire weight of the ship acts.

2. Center of Buoyancy (COB) – The point where the force of buoyancy acts.

3. Metacentric Height (GM) – The vertical distance between the COG and COB (determines stability).

4. Longitudinal Weight Distribution – Prevents hogging and sagging.

5. Transverse Weight Distribution – Avoids excessive listing.

Types of Weight Loading Conditions

• Evenly Distributed Load – Cargo or ballast spread out uniformly.

• Concentrated Load – Cargo placed at specific points (can cause instability).

• Asymmetrical Loading – Unequal weight distribution causing list.

Deck Officers must ensure correct weight placement, ballast adjustments, and cargo securing to maintain safe operations.

Beam

Beam is one of the most important structural components. Beams are usually long, straight, prismatic members and always subjected forces perpendicular to the axis of the beam.

1. Shear force

Shear force is a force, or a component of a force, that acts parallel to a plane. The shearing  force  at  any  section  of  a  beam is the algebraic sum of the lateral components of the forces acting on either side of the section.

2.Bending moment

Bending moment  is the algebraic sum of all the moments of forces acting on one side of a section through a beam.