Numericals on Force and Laws of Motion for Class 9

Numericals on Force and Laws of Motion for Class 9

Here you will find numericals on force and laws of motion for class 9.

Problem:

Find the force acting on a rocket and its acceleration if its velocity is 600 m s–1 at t = 60 s and 1000 ms^–1 at t = 80s, presuming its mass remains unchanged during this time interval and its mass is 1.67 × 10⁷ kg. The total weight of the rocket at t = 80 seconds is 1.6 × 10⁷ kg. Calculate the upward force acting on the rocket at this time caused by the burning of fuel.

Solution:

(i) Calculate the acceleration (\(a\)) and force (\(F\)) at \(t=60\) seconds: \[ \begin{aligned} a &= \frac{v-u}{t} = \frac{1000 \mathrm{~m/s} – 600 \mathrm{~m/s}}{80 \mathrm{~s} – 60 \mathrm{~s}} = \frac{400 \mathrm{~m/s}}{20 \mathrm{~s}} = 20 \mathrm{~m/s^2} \\ F &= m \cdot a = 1.67 \times 10^7 \mathrm{~kg} \times 20 \mathrm{~m/s^2} = 3.34 \times 10^8 \mathrm{~Newton} \end{aligned} \] (ii) Calculate the force (\(F\)) at \(t=80\) seconds using the new total weight: \[ F = m \cdot a = 1.6 \times 10^7 \mathrm{~kg} \times 20 \mathrm{~m/s^2} = 3.2 \times 10^8 \mathrm{~Newton} = 320 \mathrm{~MN} \]

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Problem:

A body of 10 kg initially at rest is subjected to force of 20N. Calculate the kinetic energy acquired by the body at the end of 10 seconds.

Solution:

Given: \(m = 10 \mathrm{~kg}\), \(u = 0\), \(F = 20 \mathrm{~N}\), \(t = 10 \mathrm{~s}\) First, calculate the acceleration (\(a\)) using Newton’s second law: \[ F = m \cdot a \Rightarrow a = \frac{F}{m} = \frac{20 \mathrm{~N}}{10 \mathrm{~kg}} = 2 \mathrm{~m/s^2} \] Next, calculate the final velocity (\(v\)) using the equation of motion: \[ a = \frac{v – u}{t} \Rightarrow \frac{v – 0}{10 \mathrm{~s}} = 2 \mathrm{~m/s^2} \Rightarrow v = 20 \mathrm{~m/s} \] Finally, calculate the kinetic energy (\(KE\)) using the formula for kinetic energy: \[ KE = \frac{1}{2} m v^2 = \frac{1}{2} \cdot 10 \mathrm{~kg} \cdot (20 \mathrm{~m/s})^2 = 2000 \mathrm{~J} = 2 \mathrm{~kJ} \] So, the kinetic energy acquired by the body at the end of 10 seconds is \(2 \mathrm{~kJ}\).

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Problem:

A truck starts from the rest and rolls down a hill with constant acceleration. It travels 200 m in 20 s. Find its acceleration. Find the force acting on it, if its mass is 5 metric tonnes. (1 metric tonne = 1000 kg)

Solution:

Given: \(u = 0\), \(t = 20 \mathrm{~s}\), \(s = 20 \mathrm{~m}\), \(a = ?\), \(\mathrm{F} = ?\), \(m = 5000 \mathrm{~kg}\) Using the equation of motion: \[ s = ut + \frac{1}{2}at^2 \] Substituting the given values: \[ 20 \mathrm{~m} = 0 + \frac{1}{2} \cdot a \cdot (20 \mathrm{~s})^2 \] Solving for acceleration \(a\): \[ a = \frac{400}{400} = 1 \mathrm{~m/s^2} \] Now, using Newton’s second law (\(F = ma\)) to find the force (\(\mathrm{F}\)): \[ \mathrm{F} = m \cdot a = 5000 \mathrm{~kg} \cdot 1 \mathrm{~m/s^2} = 5000 \mathrm{~Newton} \] So, the acceleration (\(a\)) is \(1 \mathrm{~m/s^2}\) and the force (\(\mathrm{F}\)) is \(5000 \mathrm{~Newton}\).

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Problem:

A bullet of mass 10 g is horizontally fired with a velocity of 100 m s–1 from pistol of mass 1 kg. What is the recoil velocity of the pistol?

Solution:

Given: \(m = 10 \mathrm{~g} = \frac{10}{1000} \mathrm{~kg}\), \(v = 100 \mathrm{~m/s}\), \(\mathrm{M} = 1 \mathrm{~kg}\), \(\mathrm{V} = ?\)
Using the formula for the velocity of the gun in the opposite direction to the bullet: \[ \mathrm{V} = -\frac{m v}{\mathrm{M}} = -\frac{\frac{10}{1000} \mathrm{~kg} \times 100 \mathrm{~m/s}}{1 \mathrm{~kg}} \] Solving for \(\mathrm{V}\): \[ \mathrm{V} = -1 \mathrm{~m/s} \] So, the velocity of the gun is \(-1 \mathrm{~m/s}\) in the opposite direction to the bullet being fired.

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Force and Laws of Motion Class 9 Important Formula

1. \(1 \, \text{Newton} = 1 \, \text{kg} \, \text{m/s}^{-2}\); newton (\(\text{N}\)) is the SI unit of force.
2. \(\vec{p} = \overrightarrow{m \mathbf{v}}\), where \(\vec{p}\) is momentum, \(m\) is mass, and \(\mathbf{v}\) is velocity.
3. \(\vec{F} = \frac{\overrightarrow{p_f} – \overrightarrow{p_i}}{t} = \frac{m \mathbf{v} – m \mathbf{u}}{t} = \frac{m(\mathbf{v} – \mathbf{u})}{t} = m \mathbf{a}\), where \(a\) = acceleration. \(\frac{v-u}{t} = a]\)
4. \(\overrightarrow{\mathbf{F}} = m \mathbf{a}\) Impulse \(= \text{force} \times \text{time} = \text{change in momentum}\)
5. \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\) : Law of conservation of momentum.
6. Recoil of gun: \(\mathbf{V} = \frac{-m \mathbf{v}}{\mathbf{M}}\), where \(m\) is the mass of the bullet, \(M\) is the mass of the gun, \(\mathbf{V}\) is the velocity of the gun, and \(\mathbf{v}\) is the velocity of the bullet.
7. \(\frac{\text{Rocket propulsion}}{\text{Velocity of rocket}} = V = \frac{-mV}{M}\), where \(m\) is the mass of burnt fuel, \(v\) is the velocity of burnt fuel, and \(M\) is the mass of the rocket at any instant.

Why are Physics Numericals Important for Class 9 Students?

1. Application of Concepts: Physics numericals allow students to apply the theoretical concepts they learn in class to real-world situations. This practical application enhances their understanding of the subject.
2. Problem-Solving Skills: Numerical problems require critical thinking and problem-solving skills. Students learn to analyze information, identify relevant formulas, and calculate solutions, skills that are valuable in many aspects of life.
3. Enhanced Understanding: By solving numerical problems, students gain a deeper insight into the underlying principles of physics. It’s one thing to memorize formulas, but it’s another to comprehend how they are derived and applied.
4. Preparation for Higher Classes: A strong foundation in physics at the Class 9 level prepares students for more advanced physics topics in higher classes. It paves the way for success in Class 10 board exams and beyond.

Tips for Solving Physics Numericals

1. Understand the Concept: Before attempting numericals, ensure you have a clear understanding of the underlying physics concepts. Review your class notes and textbook.
2. Identify Given Data: Carefully read the problem and identify the given data. This is crucial for selecting the appropriate formula.
3. Choose the Right Formula: Select the formula that best fits the problem. Familiarize yourself with the relevant equations for each topic.
4. Units and Conversions: Pay attention to units. Ensure all units are consistent throughout your calculations, and convert them if necessary.
5. Organize Your Work: Show your work step by step. This not only helps you track your progress but also allows for partial credit if you make a mistake.
6. Practice Regularly: Physics numericals improve with practice. The more problems you solve, the more confident and proficient you’ll become.
7. Seek Help When Needed: If you’re stuck on a problem, don’t hesitate to seek help from your teacher, classmates, or online resources.