PhysicsKund Physics class 9 to 12 NCERT, JEE - NEET

Notes : Ncert Class 11 Physics Chapter 5 Work , Energy and Power  1.  Scalar product or dot product of two vectors : 2. Expression for the angle between two vectors 3. Projection of a Vector Along Another Vector  4. State and Prove Work Energy Theorem  5. Work Done and it's type 6. Kinetic energy 7. Work done by a variable force 8. work-energy theorem for a variable force 9. The concept of potential energy 10.  The conservation of mechanical energy 11. The potential energy of a spring 12. Power 13. Collisions 14. Collisions-One-Dimension-1D-Collision   15. Collisions-two-Dimension-2D-Collision   16. NCERT Solution Class 11 Physics Chapter 5 Work, Energy and Power  

Collisions in Two Dimensions A two-dimensional collision occurs when two bodies collide and move in a plane after collision. Linear momentum is conserved in such collisions. Since momentum is a vector quantity, its conservation must be applied separately along the x-axis and y-axis. Consider Mass m₁ moving initially with velocity $ u$ . Mass m₂ initially at rest. After collision: m₁ moves with velocity v₁  at angle θ₁ . m₂ moves with velocity v₂  at angle θ₂ . Conservation of Linear Momentum Along the x-axis $$ m_1 u=m_1v_{1}\cos\theta_1+m_2v_{2}\cos\theta_2 $$ Along the y-axis $$ 0=m_1v_{1}\sin\theta_1-m_2v_{2}\sin\theta_2 $$ Unknown Quantities Usually, the known quantities are: $$ \{m_1,\;m_2,\;u\} $$ The unknown quantities are: $$ \{v_{1},\;v_{2},\;\theta_1,\;\theta_2\} $$ Thus, there are four unknowns but only two momentum equations. Special Case: One-Dimensional Collision If $$ \theta_1=\theta...

Collisions in One Dimension (1D Collision) Introduction A one-dimensional collision is a collision in which the motion of both bodies before and after collision takes place along the same straight line. Completely Inelastic Collision Elastic Collision 1. Completely Inelastic Collision In a completely inelastic collision, the colliding bodies stick together and move with a common velocity after collision. Given Mass m₁ moves with initial velocity u  and mass m₂ is initially at rest. \[ \theta_1=\theta_2=0 \] Conservation of Momentum \[ m_1 u=(m_1+m_2)v \] Therefore, \[ v=\frac{m_1}{m_1+m_2} u \] Final Velocity: \[ \boxed{v=\frac{m_1}{m_1+m_2} u} \] Loss of Kinetic Energy Initial kinetic energy: \[ K_i=\frac{1}{2}m_1 u^{2} \] Final kinetic energy: \[ K_f=\frac{1}{2}(m_1+m_2)v^{2} \] Loss in kinetic energy: \[ \Delta K = \frac{1}{2}m_1 u^{2} -\frac{1}{2}(m_1+m_2)v^{2} \] Substituting \[ v=\frac{m_1}{m_1+m_2} u \...

5.11 COLLISIONS Introduction In Physics, some quantities remain conserved during interactions between bodies. Two important conserved quantities are: Linear Momentum Energy Collisions provide practical applications of these conservation laws. Examples: Billiards, Carrom, Marbles, Cricket Ball striking a Bat. Collision A collision is a short-duration interaction between two bodies during which they exert large forces on each other, resulting in changes in their velocities. Collision of Two Particles Consider two particles: Mass $m_{1}$  moving with initial velocity $v_{1i}$ Mass $m_{2}$  initially at rest Before Collision $$u_1=v_{1i}$$ $$u_2=0$$ Particle m₁ moves along the positive x-axis towards particle m₂. After Collision m₁ moves with velocity v₁f making angle θ₁ with the x-axis. m₂ moves with velocity v₂f making angle θ₂ with the x-axis. The collision changes both speed and direction of motion. Conservation of Li...

Notes : Power , Definition, Unit and Dimensions - Class 11 Physics chapter 5 work energy and power 1. Definition of Power Power is the rate at which work is done or energy is transferred. $$ P=\frac{W}{t} $$ Where: \(P\) = Power \(W\) = Work done \(t\) = Time taken 2. Nature of Power Power is a scalar quantity. It has only magnitude, no direction. It can be positive, negative, or zero depending on work done. 3. Dimensional Formula of Power $$ [W]=[ML^2T^{-2}] $$ $$ P=\frac{W}{t} $$ $$ [P]=[ML^2T^{-3}] $$ 4. Units of Power SI Unit $$ 1\,W = 1\,J\,s^{-1} $$ Horse Power $$ 1\,hp = 746\,W $$ Kilowatt $$ 1\,kW = 1000\,W $$ Kilowatt-hour (Energy Unit) $$ 1\,kWh = 3.6 \times 10^6\,J $$ Note: kWh is a unit of energy, not power. 5. Types of Power Average Power $$ P_{avg}=\frac{W}{t} $$ Instantaneous Power $$ P=\vec{F}\cdot\vec{v} $$ $$ P=Fv\cos\theta $$ \(\theta = 0^\circ \Rightarrow P = Fv\) (maximum) \(\theta = 90^\circ \Ri...

Potential Energy of a Spring, Hooke's Law and Conservation of Mechanical Energy 1. Hooke's Law and Spring Force An ideal spring exerts a restoring force that is directly proportional to its displacement from the equilibrium position and acts in the opposite direction. Hooke's Law: \[ F_s=-kx \] Where: \(F_s\) = Spring force \(k\) = Spring constant \(x\) = Displacement from equilibrium position Spring Constant (k) Large \(k\) → Stiff spring Small \(k\) → Soft spring SI Unit: \[ \text{N m}^{-1} \] Meaning of the Negative Sign The negative sign indicates that the spring force always acts towards the equilibrium position. If x>0 , if Spring is Stretched and Force acts Backward. If x < 0 , if Spring is Compressed, Force acts Forward. Therefore, spring force is called a restoring force . 2. Work Done by the Spring Force Since spring force varies with displacement, it is a variable force. Therefore, work is calculated using...

Notes : Conservation of Mechanical Energy and Conservative Forces Class 11 physics chapter 5 work  The law of conservation of mechanical energy is one of the most important principles in physics. It states that when only conservative forces act on a body, the sum of its kinetic energy and potential energy remains constant throughout the motion. Mechanical Energy Mechanical energy is the total energy possessed by a body due to its motion and position. \[ E = K + V \] where, \(E\) = Mechanical Energy \(K\) = Kinetic Energy \(V\) = Potential Energy Derivation of the Law of Conservation of Mechanical Energy Step 1: Relation Between Conservative Force and Potential Energy For a conservative force, \[ F(x)=-\frac{dV}{dx} \] Multiplying both sides by a small displacement \(dx\), \[ F(x)\,dx=-\frac{dV}{dx}\,dx \] Since, \[ \frac{dV}{dx}\,dx=-dV \] Therefore, \[ F(x)\,dx=-dV \] Step 2: Work Done by the Force The infinitesimal work done ...

Notes : Concept of Potential Energy and Conservative Forces , Force-Potential Energy Relation class 11 physics chapter 5 work , energy and power 1. Potential Energy Definition Potential Energy (PE) is the energy possessed by a body due to its position or configuration. It is often called stored energy because it can be converted into other forms of energy, especially kinetic energy. Examples Water stored in a dam A stretched rubber band A compressed spring A stone held at a height above the ground Spring Analogy When a spring is compressed, work is done on it. This work gets stored inside the spring as potential energy. When the spring is released, the stored energy converts into kinetic energy. 2. Gravitational Potential Energy Consider a body of mass \(m\) raised vertically through a height \(h\) near the Earth's surface. Assumptions Height \(h\) is very small compared to Earth's radius. Acceleration due to gravity \(g\) remains constant. \[...

The Work-Energy Theorem states that the net work done on a particle is equal to the change in its kinetic energy. \[ W=\Delta K=K_f-K_i \] This theorem is valid for both constant and variable forces. Work Done by a Variable Force For a variable force \(F(x)\), the force changes with position. Therefore, the work done cannot be calculated using \[ W=F \cdot s \] Instead, work is calculated by integration: \[ W=\int_{x_i}^{x_f}F(x)\,dx \] where: \(F(x)\) = Variable force \(x_i\) = Initial position \(x_f\) = Final position Derivation of Work-Energy Theorem for a Variable Force Consider a particle of mass \(m\) moving along the x-axis under the action of a variable force \(F(x)\). Step 1: Write the Expression for Kinetic Energy Kinetic energy of the particle is: \[ K=\frac{1}{2}mv^2 \] Differentiating both sides with respect to time: \[ \frac{dK}{dt} = \frac{d}{dt} \left( \frac{1}{2}mv^2 \right) \] Since \(m\) and \(\frac{1}{2}\) are ...

In real-life situations, a force is rarely constant. Most forces change with position, time, or displacement. Such a force is called a variable force . 1. Work Done for a Small Displacement Consider a force \(F(x)\) that varies with position \(x\). If the displacement \(\Delta x\) is very small, the force can be assumed to be approximately constant over that small interval. Therefore, the small amount of work done is: \[ \Delta W = F(x)\,\Delta x \] where: \(F(x)\) = force at position \(x\) \(\Delta x\) = small displacement 2. Total Work Done To find the total work done from the initial position \(x_i\) to the final position \(x_f\), we add the work done over all small intervals: \[ W = \sum F(x)\,\Delta x \] The summation extends from \(x_i\) to \(x_f\). 3. Limiting Case As the displacement intervals become smaller and smaller, \[ \Delta x \rightarrow 0 \] the number of intervals increases without limit. In this limit, the summation becomes a definite ...

Notes : Kinetic Energy Definition, Properties, Vector Form , Units and Dimensions  Kinetic energy is one of the most important forms of mechanical energy. It is the energy possessed by a body due to its motion. Any moving object, whether it is a car, a rolling ball, or a flying airplane, possesses kinetic energy. 1. Definition of Kinetic Energy Kinetic Energy (K): The energy possessed by a body by virtue of its motion is called kinetic energy. It is equal to the amount of work done in bringing a body from rest to a given velocity. Formula: \[ K=\frac{1}{2}mv^2 \] Where: m = Mass of the body v = Velocity of the body 2. Important Properties of Kinetic Energy Kinetic energy is a scalar quantity . It depends on both mass and velocity. Kinetic energy is always positive or zero. A body at rest has zero kinetic energy. Kinetic energy depends upon the frame of reference of the observer. Kinetic energy is directly proportional to mass. Kinetic energy is direct...

Notes : Work Done : Definition, Formula, Types (Positive, Negative & Zero Work) and Units | Class 11 Notes 1. Mathematical Definition In everyday language, "work" means any physical or mental exertion. However, in physics, work has a precise mathematical definition based on force and displacement. Definition: The work done by a constant force is defined as the product of the component of the force in the direction of the displacement and the magnitude of that displacement. Vector Dot Product Formula: $W = (F \cos\theta)d = \mathbf{F} \cdot \mathbf{d}$ Variables Explained: W = Work done (It is a scalar quantity , meaning it has magnitude but no direction). F = Magnitude of the applied force. d = Magnitude of the displacement. $\theta$  = The angle between the force vector ($\mathbf{F}$) and the displacement vector ($\mathbf{d}$). 2. The Three Conditions for Zero Work ($W = 0$) According to the formula, no mechanical work is done i...

6.3.4 WORK-ENERGY THEOREM State and prove work-energy theorem for a constant force. Work-Energy Theorem : This theorem states that the work done by a force to move a body is equal to the change in kinetic energy of the body. work done by a force on body = change in kinetic energy of the body$ Proof : Consider a body of mass $m$ moving with a velocity $u$. Let a force $F$ be applied on the body, so that it is accelerated with an acceleration '$a$'. Then, $$F = ma$$ If $S$ be the distance travelled by the body during its accelerated motion in the direction of applied force, then the work done by the force $F$ on the body is given by $$W = FS = ma \, S \quad \quad \quad (\because F = ma) \quad \dots(1)$$ Let the body acquires velocity $v$ after travelling a distance $S$, then from $v^2 - u^2 = 2 \, aS$, we have $$a = \frac{v^2 - u^2}{2 \, S} \quad \dots(2)$$ Put this value in eqn. (1), we get $$W = m \left...

NCERT Solution Class 11 Physics Chapter 5 Work, Energy and Power - Physicskund  5.1 The sign of work done by a force on a body is important to understand. State carefully. if the following quantities are positive or negative: (a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket. (b) work done by gravitational force in the above case, (c) work done by friction on a body sliding down an inclined plane, (d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity, (e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest. Solution :  (a) Positive: The man applies force upwards, and the bucket moves upwards. Force and displacement are in the same direction, so the work done is positive. (b) Negative: Gravitational force acts downwards, while the bucket moves upwards. Force and displacement are in opposite directions, so the work done is negative. (c) Negative: Frictio...