Newton’s Law of Motion

• Newton’s Laws of Motion are three scientific laws that describe the relationship between a body and the forces acting upon it.
• They were first presented by Sir Isaac Newton in his work “Philosophiæ Naturalis Principia Mathematica” in 1687.

1. Newton’s first law – free body diagrams:

• “An object at rest will remain at rest, and an object in motion will continue to move with a constant velocity, unless acted upon by an external force.“
• “If an object is at rest, it will stay at rest, unless a force is applied to it”.
• “If an object is moving, it will continue to move with a constant velocity (speed and direction), unless a force is applied to it”.
• This law applies to all objects, big or small, and is a fundamental concept in understanding how objects move and respond to forces.

  Figure 1 By newton’s law of motion

• Some key points to note about the Law of Inertia:
  – Inertia is a property of an object that describes its tendency to resist changes in its motion.
  – The law applies to all objects, regardless of their mass or size.
  – The law only applies to objects that are not subject to external forces. If a force is applied, the object’s motion will change.
  – The law is often referred to as the “law of inertia” because it describes the tendency of objects to maintain their state of motion.

⇒ Free body diagram:

• A free body diagram is a graphical representation of an object and the forces acting upon it. It’s a crucial tool in physics and engineering to visualize and analyze the forces that affect an object’s motion.
• A free body diagram typically includes:
  1. The object: Represented by a box, circle, or other shape.
  2. Forces: Arrows that represent the forces acting on the object, labeled with their magnitude and direction.
  3. Axes: Coordinate axes (x, y, and sometimes z) that help to define the orientation of the forces.
     Types of forces that might be shown on a free body diagram:
     External forces:
     – Frictional force (f)
     – Normal force (N)
     – Applied force (F)
     – Gravity (g)
• Internal forces:
  – Tension (T)
  – Spring force (k)
  – Air resistance (D)
• Some benefits of using free body diagrams:
  – Help to clarify complex force systems
  – Simplify problem-solving
  – Enhance understanding of force interactions
  – Facilitate calculation of net forces and motion
  
• Figure 2 When external force applies on a moving object then it will change its motion

⇒ Examples

(1)

• After Galileo made some discoveries on friction, Newton expanded on them.
• Galileo saw balls tumbling over various curves.
  
• Figure 3 Both (starting and ending) points are equal without incline plane
• The ball will run down one side of the curve then up the other.
• Galileo noticed that if smooth surfaces were used, the ball got closer to its original height (its height at the starting point).
• Galileo reasoned that the ball would get to the original height if there were no friction.
• The ball would get to the original height if there were no outside force (unbalanced force).
  
• Figure 4 both points (ending and starting) are same but one end has an incline plane
• In all cases, the ball will return to its starting height regardless of the angles.
• The ball will go a larger distance but still not reach its initial height if the second curve’s inclination is smaller than the firsts because of friction, or the ball meeting resistance from the surface it is racing along.
• Even though the two ramps had different slopes, in the absence of friction, the ball would roll up the opposing slope to its original height.
  
• Figure 5 Second end is not equal to first starting point
• Galileo came to the conclusion that if the curve terminated without an inclination, the ball would continue indefinitely till friction finally stopped it.
• These findings led Newton to the conclusion that an item may remain in motion without the assistance of a force. The ball is really stopped from going any farther by an outside force.

(2)

• A skydiver descends to the earth with a parachute at a steady speed.
• The skydiver and parachute are in contact with nothing here, hence the free body diagram is simple (save the air).
• The weight of the skydiver and parachute, W, is balanced downward by the drag, D, pointing above.
  

• Figure 6 Skydiver falling to the ground at a constant speed.

(3)

• A climber is shown abseiling down a rock face in Figure 7 (1);
• He has just stopped to rest and is motionless.
• We have to take off the rock face in order to construct a free body diagram.
  • Figure 7 (2) illustrates the three forces at work:
  • The climber’s weight (W)
  • The rope’s tension (T)
  • He rocks face’s response (R).
• These forces act via the climber’s centre of gravity since they are immobile.
• They sum up to zero, as Figure 7(3) illustrates.
  
• Figure 7 (1) A climber is abseiling down a rock face, (2) The forces acting on the climber, (3) The forces add up to zero.

(4)

• A guy is seen ascending a ladder that leans against a smooth wall in Figure 8 (1).
• A free body diagram of the ladder with the guy standing on it is shown in Figure 8 (2).
  
• Figure 8 (1) A man climbing a ladder (2) A free body diagram for the man on the ladder.
• The forces acting on the ladder are:
  • R_W, a horizontal reaction force from the wall
  • R_F, a vertical reaction force from the floor
  • F, a horizontal frictional force from the floor
  • W, the weight of the ladder
  • R_m, a contact force from the man that is equal in size to his weight (this is not the man’s weight, which acts on him).
• Since the ladder remains stationary, the forces on it balance. So
• R_F = W + R_m
• These are the forces acting vertically
• F=R_w
• These are the forces acting horizontally

2. Newton’s second law of motion:

Newton’s Second Law, also known as the Law of Acceleration, relates the motion of an object to the force acting upon it. It states:

• “The acceleration of an object is directly proportional to the force applied and inversely proportional to its mass.”
  
• Where:
  – [math]\vec{F}[/math] is the net force acting on an object, unit is N (newton)
  – m is the mass of the object, unit is kg (kilogram)
  –  [math]\vec{a}[/math] is the acceleration of the object, unit is m/s (meter per second)
• This law means that:
  – The more force applied to an object, the more it will accelerate ([math]\vec{a} \propto \vec{F} [/math]) where the mass of an object will be constant.
  – The heavier an object is (more massive), the less it will accelerate when a force is applied([math]\vec{a} \propto \frac{1}{m}
  [/math]).
• Some key aspects of Newton’s Second Law:
  – Force and acceleration are vectors, so they have both magnitude and direction.
  – Mass is a scalar quantity, so it has only magnitude (amount of matter).
  – The law applies to all objects, big or small, and is a fundamental principle in understanding how objects move and respond to forces.
  

• Figure 9 Newton’s second Law very clearly example with respect to two different masses, different acceleration, and different force

⇒ Examples:
(1)

Figure 10 The forces on a cyclist

• Figure 9 shows the forces on a cyclist accelerating along the road. The forces in the vertical direction balance, but the force pushing her along the road, F, is greater than the drag forces, D, acting on her. The mass of the cyclist and the bicycle is 100kg. Calculate her acceleration.

Given Data:
Drag Force = [math]\vec{D} = 180 N[/math]
Road force = [math]\vec{F} = 320N[/math]
The mass of the cyclist and the bicycle= 100kg
Find data:
Acceleration =?
Formula:

[math]\vec{F} – \vec{D} = m\vec{a}[/math]

Solution:
Her acceleration can be calculated as follows.

[math]\vec{F} – \vec{D} = m\vec{a}[/math]

Put values

[math]320 -180 =100 * \vec{a}[/math]
[math]\frac{140}{100} = \vec{a} [/math]
[math]\vec{a}=1.4m/s[/math]

(2)

• According to Newton’s second rule of motion, an object’s acceleration is exactly proportional to the net force it encounters
• Force = mass * acceleration
• Changing any of those three variables will change an object’s motion. Any change in force is proportional to a change in mass and/or acceleration, and any change in mass is inversely proportional to a change in acceleration (and vice versa).
• Figure 11 By 2^nd law force on a ball applies with respect to its acceleration
• For example, if an object of mass m is replaced by an object with double the mass, 2m, you would need twice the force to move the larger mass at the same acceleration as the smaller mass. Or, of you wanted to apply the same force to both masses, the 2m mass would have to move at half the acceleration as m.
• Because both acceleration and force are vector quantities (meaning they have a specific direction) changing the direction of a force will also result in a change in the direction of acceleration.

(3)

• A passenger travels in a lift that is accelerating upwards at a rate of [math]1.5 \, \text{m/s}^2 [/math]. The passenger has a mass of 62kg. By drawing a free body diagram to show the forces acting on the passenger, calculate the reaction force that the lift exerts on her.

• Figure 12 A passenger in a lift

• Given data:
  Upward acceleration [math] = \vec{a} =1.5\,\text{m/s}^2[/math]
  Mass of the passenger = 62kg
  Force appling downward by her weight [math]= \vec{W} = m\vec{g} = 62kg*10 \, \text{m/s}^2 \\
  \vec{W}=620 N [/math]
  Find Data:
  Reaction force (upward force) =[math]\vec{F_R}= ? [/math]
  Formula:

   

  [math]\vec{F_R} – \vec{W} = m\vec{a} [/math]
  [math]\vec{F_R} = \vec{W} + m\vec{a}[/math]

  Solution:

  [math]\vec{F_R} = \vec{W} + m\vec{a}[/math]

  Put values

  [math]\vec{F_R} = 620 + 62 \times 1.5 [/math]
  [math]\vec{F_R} = 713 \, \text{N}[/math]

3.Newton’s third law of motion:

Newton’s Third Law, also known as the Law of Action and Reaction, states:
“For every action, there is an equal and opposite reaction.”

Figure 13 Action and reaction equal but in opposite direction

• This law applies to all interactions between objects, and it’s a fundamental principle in understanding how forces work.
• Some key aspects of Newton’s Third Law:
  – Forces always come in pairs (action-reaction pairs).
  – The forces are equal in magnitude and opposite in direction.
  – The law applies to all types of forces (friction, gravity, normal force, etc.).

Examples:
(1)

• When the wheel of a car turns, it pushes the road backwards. The road pushes the wheel forwards with an equal and opposite force.

Figure 14 Wheel of a car

(2)

Figure 15 Two persons apply equal forces but in opposite direction

(3)

Figure 16 If I push you with a force of 100N, you push me back with a force of 100N.

(4)

• Two balloons have been charged positively. They each experience a repulsive force from the other.
• These forces are of the same size, so each balloon (if of the same mass) is lifted through the same angle.

Figure 17 Two balloons have been same charged (positively)