In the case of a frictionless inclined table, the work done by the table surface on the ball can be analyzed through the forces acting on the ball. Since there is no friction, the only force acting parallel to the surface is gravity. Gravity does not do work against the normal force of the table. ThRead more
In the case of a frictionless inclined table, the work done by the table surface on the ball can be analyzed through the forces acting on the ball.
Since there is no friction, the only force acting parallel to the surface is gravity. Gravity does not do work against the normal force of the table. The normal force acts perpendicular to the displacement of the ball.
Work done (W) is given by the formula:
W = F • d • cos(θ)
Where:
– F is the force
– d is the displacement
– θ is the angle between the force and displacement
This implies that the displacement and the force exerted are perpendicular to each other, that is, θ = 90 degrees. This gives cos(90°) = 0. Thus the work done by the table surface on the ball is:
W = F • d • 0 = 0
Final Answer:
The work done by the table surface on the ball is zero.
To calculate the work done on the body, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. Step 1: Calculate the initial kinetic energy (K.E.₁) Since the body is initially at rest, its initial kinetic energy is: K.E.₁ = (1Read more
To calculate the work done on the body, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
Step 1: Calculate the initial kinetic energy (K.E.₁)
Since the body is initially at rest, its initial kinetic energy is:
K.E.₁ = (1/2) m v₁²
K.E.₁ = (1/2) × 10 kg × (0 m/s)²
K.E.₁ = 0 J
Step 2: Final kinetic energy K.E.₂
When the body has achieved a velocity of 10 m/s, the final kinetic energy is:
We can find the maximum height reached by a ball after it is dropped from height h and bounces on the ground with a coefficient of restitution e using the following analysis: 1. When the ball is dropped from height h, it gains kinetic energy just before hitting the ground. The potential energy at heRead more
We can find the maximum height reached by a ball after it is dropped from height h and bounces on the ground with a coefficient of restitution e using the following analysis:
1. When the ball is dropped from height h, it gains kinetic energy just before hitting the ground. The potential energy at height h is converted to kinetic energy:
Potential Energy (PE) = mgh
Kinetic Energy (KE) at the moment of impact = mgh
2. When it bounces, some energy is lost due to the coefficient of restitution e. The coefficient of restitution is defined as the ratio of the velocity after the bounce to the velocity before the bounce:
e = (velocity after bounce) / (velocity before bounce)
3. Velocity at the instant of hitting the ground (v) is determined by the relation:
v = √(2gh)
4. Just after the bounce, the velocity is given by,
velocity after bounce = e * v = e * √(2gh)
5. Maximum height (h’) attained just after the bounce can be calculated from the K.E. at the instant of bounce
K.E. after bounce = (1/2) m (e * √(2gh))²
This kinetic energy is turned back into potential energy at the maximum height:
PE at max height = mgh’
Thus, (1/2) m (e² * 2gh) = mgh’
Simplifying gives:
h’ = e²h
Final Answer:
The maximum height after the bounce is e²h.
To find the velocity of two particles that collide and stick together, we can use the principles of conservation of momentum. Let: - Mass of each particle = m - Initial velocity of the first particle (moving north) = v - Initial velocity of the second particle (moving east) = v 1. Momentum Before CoRead more
To find the velocity of two particles that collide and stick together, we can use the principles of conservation of momentum.
Let:
– Mass of each particle = m
– Initial velocity of the first particle (moving north) = v
– Initial velocity of the second particle (moving east) = v
1. Momentum Before Collision:
– Momentum of the first particle (north): p₁ = m * v
– Momentum of the second particle (east): p₂ = m * v
2. Total Momentum Before Collision:
– The momentum vector of the first particle is (0, mv) (north direction).
– The momentum vector of the second particle is (mv, 0) (east direction).
– Therefore, the total momentum vector before the collision is:
P_initial = (mv, mv)
3. Momentum After Collision:
– Since the two particles stick together after the collision, the combined mass is 2m.
– Let the velocity of the combined mass after the collision be V, and its direction will be towards the northeast.
4. Using Pythagoras’ Theorem:
– The magnitude of the momentum vector after the collision can be found using:
P_final = √[(mv)² + (mv)²] = √[2(mv)²] = mv√2
5. Calculating the Final Velocity:
– The total momentum after the collision is equal to the momentum before the collision:
P_final = 2m * V
– Setting them equal:
mv√2 = 2m * V
– Canceling m from both sides:
v√2 = 2V
– Solving for V:
V = (v√2) / 2 = v / √2
Final Answer:
The velocity of the combined mass after the collision is v/√2.
To calculate the work done by the variable force F = x + x³ over the displacement from x = 0 m to x = 2 m, we use the work integral: Work W = ∫[0 to 2] F(x) dx Where: - F(x) = x + x³ - x₁ = 0, x₂ = 2 Step 1: Set up the integral W = ∫[0 to 2] (x + x³) dx Step 2: Split and evaluate the integral W = ∫[Read more
To calculate the work done by the variable force F = x + x³ over the displacement from x = 0 m to x = 2 m, we use the work integral:
Work W = ∫[0 to 2] F(x) dx
Where:
– F(x) = x + x³
– x₁ = 0, x₂ = 2
Step 1: Set up the integral
W = ∫[0 to 2] (x + x³) dx
Step 2: Split and evaluate the integral
W = ∫[0 to 2] x dx + ∫[0 to 2] x³ dx
1. For ∫[0 to 2] x dx:
∫ x dx = (1/2) x² |[0 to 2] = (1/2) (2²) – (1/2) (0²) = 2
2. For ∫[0 to 2] x³ dx:
∫ x³ dx = (1/4) x⁴ |[0 to 2] = (1/4) (2⁴) – (1/4) (0⁴) = (16/4) – 0 = 4
Step 3: Add the results
W = 2 + 4 = 6 J
Final Answer:
The work done by the variable force is 6 J.
The coefficient of restitution, e, is defined as the ratio of the relative velocity of separation to the relative velocity of approach between two colliding objects. For a perfectly elastic collision, the kinetic energy and momentum are conserved, and the objects rebound without any loss of energy.Read more
The coefficient of restitution, e, is defined as the ratio of the relative velocity of separation to the relative velocity of approach between two colliding objects.
For a perfectly elastic collision, the kinetic energy and momentum are conserved, and the objects rebound without any loss of energy. Therefore, the coefficient of restitution for a perfectly elastic collision is:
e = 1
Final Answer:
The coefficient of restitution, e, for a perfectly elastic collision is 1.
Conservative forces are those for which the work done on an object depends only upon the initial and final positions of the object, not on the route taken between them. That property guarantees that when an object is moved under the influence of a conservative force and returns to its original positRead more
Conservative forces are those for which the work done on an object depends only upon the initial and final positions of the object, not on the route taken between them. That property guarantees that when an object is moved under the influence of a conservative force and returns to its original position, the total work done by the force is zero.
One of the most important characteristics of conservative forces is that they are associated with potential energy. When an object moves in a conservative force field, its potential energy changes, and this change is equal to the work done by the force. Thus, for example, in the gravitational field, when an object is lifted, it gains potential energy equal to the work done against gravity.
In addition, conservative forces are necessary in the principle of energy conservation. The total mechanical energy—potential and kinetic energy—is conserved in systems affected by these forces. Energy can be transformed from potential to kinetic form and vice versa; however, its total is kept constant.
Examples of conservative forces include gravitational force, electrostatic force, and spring force. These forces are fundamental in classical mechanics, explaining various physical phenomena and the behavior of systems in motion and energy transfer.
To derive the kinetic energy expression, we begin with the work-energy relationship. The work-energy principle states that work done on an object is equal to the change in its kinetic energy. Work done when a force acts on an object can be expressed as force times distance, causing the body to moveRead more
To derive the kinetic energy expression, we begin with the work-energy relationship. The work-energy principle states that work done on an object is equal to the change in its kinetic energy. Work done when a force acts on an object can be expressed as force times distance, causing the body to move some distance in the direction of the applied force.
By Newton’s second law, force is defined as the product of mass and acceleration. If a body has acceleration, there is a change in its velocity. If the body starts from rest and the acceleration is uniform, its final velocity is determined by the distance and the acceleration of the body.
Putting the definition of force and distance into the equation for work done, we arrive at the work done on the object being related to its mass and the square of its velocity. The work done to accelerate an object from rest to some velocity goes directly into its kinetic energy.
Therefore, kinetic energy is defined as the energy an object has due to its motion, dependent on its mass and the square of its velocity. Therein, the interrelation of both mass and speed shows how important those values are in establishing the energy belonging to moving objects.
Kinetic energy is the energy an object has due to its motion. It depends on two main things: the mass of the object and the velocity at which the object moves. In other words, the more massive an object is or the greater the velocity, the higher the kinetic energy. This sort of energy is a scalar quRead more
Kinetic energy is the energy an object has due to its motion. It depends on two main things: the mass of the object and the velocity at which the object moves. In other words, the more massive an object is or the greater the velocity, the higher the kinetic energy. This sort of energy is a scalar quantity—a value with magnitude but no direction.
There are many examples of kinetic energy in real life: for example, a moving car, which increases in kinetic energy as the speed increases or even as the mass of the object increases. Therefore, a truck moving at a speed similar to that of a bicycle holds more kinetic energy due to its increased mass. In sports, when a soccer player kicks a ball, there is kinetic energy in the ball as it moves, which affects its speed and direction.
Flowing water in rivers holds kinetic energy that can be used in hydroelectric power. An airplane in flight contains plenty of kinetic energy, which is needed to maintain lift and propulsion. Even running animals—such as dogs in a park—are examples of kinetic energy, as they move fast.
In a nutshell, kinetic energy lies at the basis of several physical phenomena, apparently experienced in everyday situations, and this forms one of the core ideas regarding motion and the transfer of energies.
Einstein's mass-energy equivalence shows that mass and energy are inextricably linked and interchangeable. This means that a small quantity of mass can be converted into a huge amount of energy because the speed of light squared is very large. That was the revolutionary idea that transformed our perRead more
Einstein’s mass-energy equivalence shows that mass and energy are inextricably linked and interchangeable. This means that a small quantity of mass can be converted into a huge amount of energy because the speed of light squared is very large. That was the revolutionary idea that transformed our perception of physics, since it showed that mass could be interpreted as some sort of stored energy.
Practical applications of mass-energy equivalence include:
1. Nuclear Energy: A very small amount of mass gets converted to energy in nuclear reactions like fission and fusion. This powers nuclear reactors and accounts for the explosive energy released by atomic bombs.
2. Medical Applications: Many techniques, such as Positron Emission Tomography (PET) scans, depend on mass-energy equivalence. In a PET scan, positrons (the antimatter equivalent of electrons) annihilate with electrons to produce gamma rays that form images of metabolic activity in the body.
3. Particle Physics: In high-energy particle colliders, particles are accelerated to nearly the speed of light. When these particles collide, mass can be converted into energy, helping scientists discover new particles and understand fundamental forces.
4. Astrophysics: The principle can be applied to explain processes in stars, whereby nuclear fusion changes hydrogen into helium, releasing enormous energy that powers stars like the Sun.
Einstein’s mass-energy equivalence has had a tremendous impact across different fields, shaping energy production, medical imaging, and fundamental research in physics.
A ball moves in a frictionless inclined table without slipping. The work done by the table surface on the ball is
In the case of a frictionless inclined table, the work done by the table surface on the ball can be analyzed through the forces acting on the ball. Since there is no friction, the only force acting parallel to the surface is gravity. Gravity does not do work against the normal force of the table. ThRead more
In the case of a frictionless inclined table, the work done by the table surface on the ball can be analyzed through the forces acting on the ball.
Since there is no friction, the only force acting parallel to the surface is gravity. Gravity does not do work against the normal force of the table. The normal force acts perpendicular to the displacement of the ball.
Work done (W) is given by the formula:
W = F • d • cos(θ)
Where:
– F is the force
– d is the displacement
– θ is the angle between the force and displacement
This implies that the displacement and the force exerted are perpendicular to each other, that is, θ = 90 degrees. This gives cos(90°) = 0. Thus the work done by the table surface on the ball is:
W = F • d • 0 = 0
Final Answer:
The work done by the table surface on the ball is zero.
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A body of mass 10 kg initially at rest acquires velocity of 10 ms⁻¹. What is the work done?
To calculate the work done on the body, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. Step 1: Calculate the initial kinetic energy (K.E.₁) Since the body is initially at rest, its initial kinetic energy is: K.E.₁ = (1Read more
To calculate the work done on the body, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
Step 1: Calculate the initial kinetic energy (K.E.₁)
Since the body is initially at rest, its initial kinetic energy is:
K.E.₁ = (1/2) m v₁²
K.E.₁ = (1/2) × 10 kg × (0 m/s)²
K.E.₁ = 0 J
Step 2: Final kinetic energy K.E.₂
When the body has achieved a velocity of 10 m/s, the final kinetic energy is:
K.E.₂ = (1/2) m v₂²
K.E.₂ = (1/2) × 10 kg × (10 m/s)²
K.E.₂ = (1/2) × 10 × 100
K.E.₂ = 500 J
Step 3: Work done (W)
The work done equals the change in kinetic energy
W = K.E.₂ – K.E.₁
W = 500 J – 0 J
W = 500 J
Final Answer:
Work done is 500 J.
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A ball is dropped from height h on the ground where coefficient of restitution is e. After one balance the maximum height is
We can find the maximum height reached by a ball after it is dropped from height h and bounces on the ground with a coefficient of restitution e using the following analysis: 1. When the ball is dropped from height h, it gains kinetic energy just before hitting the ground. The potential energy at heRead more
We can find the maximum height reached by a ball after it is dropped from height h and bounces on the ground with a coefficient of restitution e using the following analysis:
1. When the ball is dropped from height h, it gains kinetic energy just before hitting the ground. The potential energy at height h is converted to kinetic energy:
Potential Energy (PE) = mgh
Kinetic Energy (KE) at the moment of impact = mgh
2. When it bounces, some energy is lost due to the coefficient of restitution e. The coefficient of restitution is defined as the ratio of the velocity after the bounce to the velocity before the bounce:
e = (velocity after bounce) / (velocity before bounce)
3. Velocity at the instant of hitting the ground (v) is determined by the relation:
v = √(2gh)
4. Just after the bounce, the velocity is given by,
velocity after bounce = e * v = e * √(2gh)
5. Maximum height (h’) attained just after the bounce can be calculated from the K.E. at the instant of bounce
K.E. after bounce = (1/2) m (e * √(2gh))²
This kinetic energy is turned back into potential energy at the maximum height:
PE at max height = mgh’
Thus, (1/2) m (e² * 2gh) = mgh’
Simplifying gives:
h’ = e²h
Final Answer:
The maximum height after the bounce is e²h.
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A particle of mass m having velocity v moving towards north collides with similar particle moving with same velocity towards east. The two particles stick together and move towards north east with a velocity
To find the velocity of two particles that collide and stick together, we can use the principles of conservation of momentum. Let: - Mass of each particle = m - Initial velocity of the first particle (moving north) = v - Initial velocity of the second particle (moving east) = v 1. Momentum Before CoRead more
To find the velocity of two particles that collide and stick together, we can use the principles of conservation of momentum.
Let:
– Mass of each particle = m
– Initial velocity of the first particle (moving north) = v
– Initial velocity of the second particle (moving east) = v
1. Momentum Before Collision:
– Momentum of the first particle (north): p₁ = m * v
– Momentum of the second particle (east): p₂ = m * v
2. Total Momentum Before Collision:
– The momentum vector of the first particle is (0, mv) (north direction).
– The momentum vector of the second particle is (mv, 0) (east direction).
– Therefore, the total momentum vector before the collision is:
P_initial = (mv, mv)
3. Momentum After Collision:
– Since the two particles stick together after the collision, the combined mass is 2m.
– Let the velocity of the combined mass after the collision be V, and its direction will be towards the northeast.
4. Using Pythagoras’ Theorem:
– The magnitude of the momentum vector after the collision can be found using:
P_final = √[(mv)² + (mv)²] = √[2(mv)²] = mv√2
5. Calculating the Final Velocity:
– The total momentum after the collision is equal to the momentum before the collision:
P_final = 2m * V
– Setting them equal:
mv√2 = 2m * V
– Canceling m from both sides:
v√2 = 2V
– Solving for V:
V = (v√2) / 2 = v / √2
Final Answer:
See lessThe velocity of the combined mass after the collision is v/√2.
The work done by an applied variable force F = x + x³ from x = 0 m to x = 2 m, where x is displacement , is
To calculate the work done by the variable force F = x + x³ over the displacement from x = 0 m to x = 2 m, we use the work integral: Work W = ∫[0 to 2] F(x) dx Where: - F(x) = x + x³ - x₁ = 0, x₂ = 2 Step 1: Set up the integral W = ∫[0 to 2] (x + x³) dx Step 2: Split and evaluate the integral W = ∫[Read more
To calculate the work done by the variable force F = x + x³ over the displacement from x = 0 m to x = 2 m, we use the work integral:
Work W = ∫[0 to 2] F(x) dx
Where:
– F(x) = x + x³
– x₁ = 0, x₂ = 2
Step 1: Set up the integral
W = ∫[0 to 2] (x + x³) dx
Step 2: Split and evaluate the integral
W = ∫[0 to 2] x dx + ∫[0 to 2] x³ dx
1. For ∫[0 to 2] x dx:
∫ x dx = (1/2) x² |[0 to 2] = (1/2) (2²) – (1/2) (0²) = 2
2. For ∫[0 to 2] x³ dx:
∫ x³ dx = (1/4) x⁴ |[0 to 2] = (1/4) (2⁴) – (1/4) (0⁴) = (16/4) – 0 = 4
Step 3: Add the results
W = 2 + 4 = 6 J
Final Answer:
The work done by the variable force is 6 J.
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The coefficient of restitute, e, for a perfectly elastic collision is?
The coefficient of restitution, e, is defined as the ratio of the relative velocity of separation to the relative velocity of approach between two colliding objects. For a perfectly elastic collision, the kinetic energy and momentum are conserved, and the objects rebound without any loss of energy.Read more
The coefficient of restitution, e, is defined as the ratio of the relative velocity of separation to the relative velocity of approach between two colliding objects.
For a perfectly elastic collision, the kinetic energy and momentum are conserved, and the objects rebound without any loss of energy. Therefore, the coefficient of restitution for a perfectly elastic collision is:
e = 1
Final Answer:
The coefficient of restitution, e, for a perfectly elastic collision is 1.
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What are conservative forces? Explains
Conservative forces are those for which the work done on an object depends only upon the initial and final positions of the object, not on the route taken between them. That property guarantees that when an object is moved under the influence of a conservative force and returns to its original positRead more
Conservative forces are those for which the work done on an object depends only upon the initial and final positions of the object, not on the route taken between them. That property guarantees that when an object is moved under the influence of a conservative force and returns to its original position, the total work done by the force is zero.
One of the most important characteristics of conservative forces is that they are associated with potential energy. When an object moves in a conservative force field, its potential energy changes, and this change is equal to the work done by the force. Thus, for example, in the gravitational field, when an object is lifted, it gains potential energy equal to the work done against gravity.
In addition, conservative forces are necessary in the principle of energy conservation. The total mechanical energy—potential and kinetic energy—is conserved in systems affected by these forces. Energy can be transformed from potential to kinetic form and vice versa; however, its total is kept constant.
Examples of conservative forces include gravitational force, electrostatic force, and spring force. These forces are fundamental in classical mechanics, explaining various physical phenomena and the behavior of systems in motion and energy transfer.
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Derive an expression for the kinetic energy of a body of mass m moving with velocity v.
To derive the kinetic energy expression, we begin with the work-energy relationship. The work-energy principle states that work done on an object is equal to the change in its kinetic energy. Work done when a force acts on an object can be expressed as force times distance, causing the body to moveRead more
To derive the kinetic energy expression, we begin with the work-energy relationship. The work-energy principle states that work done on an object is equal to the change in its kinetic energy. Work done when a force acts on an object can be expressed as force times distance, causing the body to move some distance in the direction of the applied force.
By Newton’s second law, force is defined as the product of mass and acceleration. If a body has acceleration, there is a change in its velocity. If the body starts from rest and the acceleration is uniform, its final velocity is determined by the distance and the acceleration of the body.
Putting the definition of force and distance into the equation for work done, we arrive at the work done on the object being related to its mass and the square of its velocity. The work done to accelerate an object from rest to some velocity goes directly into its kinetic energy.
Therefore, kinetic energy is defined as the energy an object has due to its motion, dependent on its mass and the square of its velocity. Therein, the interrelation of both mass and speed shows how important those values are in establishing the energy belonging to moving objects.
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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-5/
What is kinetic energy? Give some examples.
Kinetic energy is the energy an object has due to its motion. It depends on two main things: the mass of the object and the velocity at which the object moves. In other words, the more massive an object is or the greater the velocity, the higher the kinetic energy. This sort of energy is a scalar quRead more
Kinetic energy is the energy an object has due to its motion. It depends on two main things: the mass of the object and the velocity at which the object moves. In other words, the more massive an object is or the greater the velocity, the higher the kinetic energy. This sort of energy is a scalar quantity—a value with magnitude but no direction.
There are many examples of kinetic energy in real life: for example, a moving car, which increases in kinetic energy as the speed increases or even as the mass of the object increases. Therefore, a truck moving at a speed similar to that of a bicycle holds more kinetic energy due to its increased mass. In sports, when a soccer player kicks a ball, there is kinetic energy in the ball as it moves, which affects its speed and direction.
Flowing water in rivers holds kinetic energy that can be used in hydroelectric power. An airplane in flight contains plenty of kinetic energy, which is needed to maintain lift and propulsion. Even running animals—such as dogs in a park—are examples of kinetic energy, as they move fast.
In a nutshell, kinetic energy lies at the basis of several physical phenomena, apparently experienced in everyday situations, and this forms one of the core ideas regarding motion and the transfer of energies.
Click here:
See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-5/
What is Einstein’s mass-energy equivalence? Mention some of its practical applications.
Einstein's mass-energy equivalence shows that mass and energy are inextricably linked and interchangeable. This means that a small quantity of mass can be converted into a huge amount of energy because the speed of light squared is very large. That was the revolutionary idea that transformed our perRead more
Einstein’s mass-energy equivalence shows that mass and energy are inextricably linked and interchangeable. This means that a small quantity of mass can be converted into a huge amount of energy because the speed of light squared is very large. That was the revolutionary idea that transformed our perception of physics, since it showed that mass could be interpreted as some sort of stored energy.
Practical applications of mass-energy equivalence include:
1. Nuclear Energy: A very small amount of mass gets converted to energy in nuclear reactions like fission and fusion. This powers nuclear reactors and accounts for the explosive energy released by atomic bombs.
2. Medical Applications: Many techniques, such as Positron Emission Tomography (PET) scans, depend on mass-energy equivalence. In a PET scan, positrons (the antimatter equivalent of electrons) annihilate with electrons to produce gamma rays that form images of metabolic activity in the body.
3. Particle Physics: In high-energy particle colliders, particles are accelerated to nearly the speed of light. When these particles collide, mass can be converted into energy, helping scientists discover new particles and understand fundamental forces.
4. Astrophysics: The principle can be applied to explain processes in stars, whereby nuclear fusion changes hydrogen into helium, releasing enormous energy that powers stars like the Sun.
Einstein’s mass-energy equivalence has had a tremendous impact across different fields, shaping energy production, medical imaging, and fundamental research in physics.
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See lesshttps://www.tiwariacademy.com/ncert-solutions/class-11/physics/chapter-5/