Mastering Newton's law of motion with theory, examples and problems.
Newton’s Laws of Motion: The Science Behind Movement
Estimated reading time: 9 minutes
Sir Isaac Newton’s three laws of motion provide a fundamental framework for not only understanding how objects move but also how they interact with forces. These principles not only explain everyday activities but also govern large-scale astronomical motions. By diving deeper into these laws, including the vector nature of force, we can gain a clearer understanding of their significance.
Newton’s Laws of Motion Explained
The First Law: The Law of Inertia
Newton’s first law states that:
An object will remain at rest or in uniform motion unless acted upon by an external force.
This means:
- A stationary object will not move unless a force is applied.
- A moving object continues at the same velocity unless an external force causes a change.
Real-World Applications of Inertia
- Car Safety: Seat belts are crucial in preventing passengers from continuing in motion during sudden stops.
- Tablecloth Trick: A quick pull on a tablecloth leaves dishes undisturbed due to their inertia.
- Space Travel: In space, objects continue moving indefinitely due to a lack of frictional forces.
The Second Law: The Law of Acceleration
Newton’s second law quantifies force using the equation:
The force applied to an object is equal to the product of its mass and acceleration.
![\[\mathbf{F}=m\cdot \mathbf{a}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-f9c45378e1d549f42392125b6159fd6b_l3.png "Rendered by QuickLaTeX.com")
where:
is force (in Newtons, 
),
is mass (in kilograms, 
),
is acceleration (in meters per second squared, 
).
This law explains why heavier objects require more force to accelerate than lighter ones.
Practical Applications of the Second Law
- Sports Performance: A heavier football requires more force to kick it the same distance as a lighter one.
- Rocket Launches: Large thrust is needed to propel a rocket due to its significant mass.
- Braking in Cars: The stopping distance increases with greater speed due to larger forces involved.
The Third Law: Action and Reaction
Newton’s third law states:
For every action, there is an equal and opposite reaction.
This means that:
![\[F_{\text{action}}= -F_{\text{reaction}}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-0c6c45931a57ca651406b83b666f5ebf_l3.png "Rendered by QuickLaTeX.com")
Examples of Newton’s Third Law
- Rowing a Boat: The oars push water backward, and the water pushes the boat forward.
- Recoil of a Gun: When a bullet is fired forward, the gun moves backward with equal force.
- Jumping Off a Boat: Pushing backward results in the boat moving away while the person jumps forward.
The Vector Nature of Force
Forces are vector quantities, meaning they have both magnitude and direction. The net force acting on an object depends on both the strength and direction of the applied forces.
Vector Addition of Forces
When multiple forces act on an object, the resultant force is the vector sum of all individual forces. If forces act along a straight line, they can be added algebraically. However, if they act at angles, vector addition (using components or graphical methods) must be used.
For example, if two forces of 
and 
act at a right angle, the resultant force 
is given by:
![\[F_r = \sqrt{(5^{2} + 10^{2})} = \sqrt{125} \approx 11.18 N\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-ff8c057794a6babc334baa56770caf54_l3.png "Rendered by QuickLaTeX.com")
Resolving Forces into Components
A force 
applied at an angle 
can be broken down into:
![\[F_x = F \cdot \cos {\theta}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-384e913191b1a07ceba904c8638b5377_l3.png "Rendered by QuickLaTeX.com")
![\[F_y = F\cdot \sin {\theta$}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-4c0b8ce5c258f7f3757022eb76f3fa2a_l3.png "Rendered by QuickLaTeX.com")
This is useful in analyzing inclined planes, projectile motion, and equilibrium problems.
Newton’s Laws and Planetary Motion
Newton’s laws also explain celestial mechanics. Planets move in elliptical orbits due to the gravitational force exerted by the Sun.
- The first law describes how planets continue moving unless acted upon by an external force.
- The second law explains the acceleration of planets due to the Sun’s gravitational pull.
- The third law applies to gravitational interactions: the Sun pulls on planets, and planets pull on the Sun with equal force.
The Connection Between Newton’s Laws and Kepler’s Laws
Newton’s second law and the law of universal gravitation lead to Kepler’s laws of planetary motion, which describe how planets orbit the Sun.
Common Misconceptions About Newton’s Laws
Misunderstandings About the First Law
Myth:
An object in motion will eventually stop on its own.
Fact:
An object stops due to friction or another force, not on its own.
Misconceptions About the Second Law
Myth:
An object needs a force to stay in motion.
Fact:
An object only needs a force to change its motion, not to sustain it.
Misinformation Regarding the Third Law
Myth:
Action and reaction forces cancel out.
Fact:
They act on different objects, so they do not cancel each other.
Numerical Problems
Problem 1: Finding Acceleration
A car of mass 1500 kg experiences a net force of 6000 N. What is its acceleration?
Solution:
![\[F = ma \Rightarrow a = \frac{F}{m} = \frac{6000}{1500} = 4 \text{ m s}^{-2}\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-4a69fb7425e99308630dd4862b0260ac_l3.png "Rendered by QuickLaTeX.com")
Therefore, 
.
Problem 2: Force Acting at an Angle
A force of 50 N is applied at an angle of 30° to the horizontal. Find the horizontal and vertical components of the force.
Solution:
![\[F_x = 50 \cos 30° \approx 43.3 N, \quad F_y = 50 \sin 30° = 25 N\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-37b106d935c54baaf6db36c4e3c1c331_l3.png "Rendered by QuickLaTeX.com")
Therefore, 25N.
Problem 3: Newton’s Third Law Application
A swimmer pushes water backward with a force of 80 N. What force does the water exert on the swimmer?
Solution:
By Newton’s third law, the water exerts an equal and opposite force of 80 N forward.
Problem 4: Multiple Forces on an Object
A box is subjected to two forces: 20 N to the right and 30 N at an angle of 60° above the horizontal. Find the resultant force.
Solution:
Resolving the 30 N force into components:
![\[F_x = 30 \cos 60° = 15 N, \quad F_y = 30 \sin 60° \approx 25.98 N\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-d339eb7323a6aa9d5282388e0af862cb_l3.png "Rendered by QuickLaTeX.com")
Total horizontal force:
![\[F_{total,x} = 20 + 15 = 35 N\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-e5f80ba1aa46dd1d99c460ee3e984b6e_l3.png "Rendered by QuickLaTeX.com")
Subsequently, the total vertical force:
![\[F_{total,y} = 25.98 N\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-be5a9db71c478e0fa1f34b96c95f5727_l3.png "Rendered by QuickLaTeX.com")
Therefore, the resultant force:
![\[F_r = \sqrt{(35^2 + 25.98^2)} \approx 43.5 N\]](https://entechonline.com/wp-content/ql-cache/quicklatex.com-c1cbb10b7c24f27728cd006092ee14ff_l3.png "Rendered by QuickLaTeX.com")
Conclusion
In essence, Newton’s laws of motion govern everyday life and astronomical phenomena alike. Meaning that, understanding the vector nature of force enhances our comprehension of how forces interact and influence motion. Therefore, these principles are fundamental to physics, engineering, and space exploration.
FAQs
Q: What are Newton’s 3 laws of motion?
A: Newton’s 3 laws of motion, formulated by Sir Isaac Newton, are fundamental principles that describe the relationship between the motion of an object and the forces acting on it. Consequently, they include the first law (law of inertia), the second law (force equals mass times acceleration), and the third law (law of action and reaction).
Q: What does Newton’s 1st law of motion state?
A: Newton’s first law of motion states that an object will remain at rest or in uniform motion in a straight line unless acted upon by a force. Hence, this law emphasizes the concept of inertia, which is the tendency of an object to resist changes in its state of motion.
Q: How does Newton’s 2nd law of motion relate to force?
A: Newton’s second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Subsequently, this relationship can be expressed with the formula 
, where is the force, is the mass, and 
is the acceleration.
Q: What is the significance of Newton’s 3rd law of motion?
A: Newton’s third law of motion states that for every action, there is an equal and opposite reaction. This means that whenever one object exerts a force on another, the second object exerts a force of equal magnitude but in the opposite direction on the first object.
Q: Can you give an example of the law of action and reaction?
A: A common example of the law of action and reaction is seen when you push against a wall. When you apply a force to the wall (action), the wall exerts an equal and opposite force back on you (reaction), which is why you do not move through the wall.
Q: How do Newton’s laws of motion apply to everyday activities?
A: Newton’s laws of motion apply to a wide range of everyday activities, from driving a car to playing sports. For instance, when you accelerate a car, you are applying a force (according to Newton’s second law), and when you jump off a diving board, you push down (action) and the board pushes you up (reaction) according to Newton’s third law.
Q: What is inertia according to Newton’s 1st law?
A: Inertia is the property of an object to maintain its state of motion. According to Newton’s 1st law, an object will not change its motion unless acted upon by a force. This means that a stationary object will remain at rest, and a moving object will continue to move in a straight line at a constant speed unless influenced by an external force.
Q: How does mass affect acceleration in Newton’s 2nd law?
A: According to Newton’s 2nd law of motion, the mass of an object affects its acceleration when a force is applied. Hence, a larger mass will require a greater amount of force to achieve the same acceleration as a smaller mass. This illustrates the inverse relationship between mass and acceleration.
Q: What does the 3rd law of motion imply about interactions between objects?
A: The 3rd law of motion implies that all interactions between objects involve forces that are equal in magnitude and opposite in direction. This means that when one object exerts a force on another, the second object responds with a force of the same strength but in the opposite direction, highlighting the interconnectedness of forces in motion.
References
- Antippa, A. F. (2003). Unification of Newton’s laws of motion. Canadian Journal of Physics, 81(5), 713–735. https://doi.org/10.1139/p03-081
- Dunstan, D. J. (2008). Derivation of special relativity from Maxwell and Newton. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366(1871), 1861–1865. https://doi.org/10.1098/rsta.2007.2195
- Hsu, L. (2001). Teaching Newton’s laws before projectile motion. The Physics Teacher, 39(4), 206–209. https://doi.org/10.1119/1.1367784
- Katsikadelis, J. T. (2018). Derivation of Newton’s law of motion from Kepler’s laws of planetary motion. Archive of Applied Mechanics, 88(1-2), 27–38. https://doi.org/10.1007/s00419-017-1245-x
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