Work Energy And Power

Step 1: Concept
Potential energy .
Step 2: Analysis
At the mean position, the displacement .
Step 3: Conclusion
Substituting into the formula, .
Final Answer: (A)

Step 1: Concept
Escape velocity .
Step 2: Analysis
.
Step 3: Conclusion
The escape velocity remains the same as .
Final Answer: (A)

Step 1: Concept
Power .
Step 2: Analysis
. .
Step 3: Conclusion
. Since 1 kg of water 1 liter, the amount is 1200 liters.
Final Answer: (C)

Step 1: Concept
Velocity at infinity .
Step 2: Analysis
Given . .
Step 3: Conclusion
km/s.
Final Answer: (B)

Step 1: Property of K
The equilibrium constant ( ) is a characteristic constant for a given reaction.
Step 2: Dependency
depends only on the temperature for a specific chemical equation.
Step 3: Conclusion
Adding more reactants or products changes the equilibrium position but does not change the value of the equilibrium constant itself. It remains 6.4.
Final Answer: (a)

Step 1: First Factor
Using for small : .
Step 2: Second Factor
.
Step 3: Combining
.
Final Answer: (a)

Step 1: Simplifying Expression
.
Step 2: Symmetry Analysis
.
Step 3: Coefficients
In the expansion, only even powers of exist. This means all odd-indexed coefficients ( ) are zero.
Step 4: Result
Since , the sum .
Final Answer: (d)

Step 1: Exothermic Nature
Physical adsorption (physisorption) is an exothermic process.
Step 2: Le Chatelier's Principle
According to Le Chatelier's Principle, lower temperatures favor exothermic processes.
Step 3: Result
Therefore, the rate of physical adsorption increases as the temperature decreases.
Final Answer: (b)

Step 1: Expression for Force.
The force on the particle is related to the rate of change of momentum: For the -component of momentum , we have: For the -component of momentum , we have: Thus, the force components are:

Step 2: Calculate the angle between and .
The angle between the vectors and is given by the formula: Substitute the values for and : Simplifying: This implies .

Step 1: Understand the energy in the spring-mass system.
In a spring-mass system, the maximum kinetic energy is equal to the maximum potential energy stored in the spring when it is stretched or compressed. The potential energy in the spring is given by: where: - is the spring constant, and - is the displacement from the equilibrium position.

Step 2: Calculate the potential energy.
Here, the spring constant , and the displacement . Substituting the values into the formula for potential energy:

Step 3: Conclusion.
The maximum kinetic energy of the system is , which is option (1).

Step 1: Use the power formula.
The power transferred to the turbine is the rate of change of potential energy, given by: where , , .

Step 2: Substitute the values.
Substituting the known values into the formula:

Step 3: Conclusion.
The power transferred to the turbine is 100 kW, which is option (1).

Step 1: Use the escape velocity formula.
The energy required for escape is given by the gravitational potential energy required to move the body from the surface to infinity: where,
,
, ,
and .

Step 2: Substitute values.
Substitute the known values into the formula:

Step 3: Conclusion.
The required energy is approximately , which is option (3).

Step 1: Use the impulse-momentum theorem.
The change in momentum ( ) is equal to the impulse applied to the object. Impulse is given by: where: - is the force applied, - is the time for which the force acts.

Step 2: Substitute the given values.
Given that and , we can substitute into the formula:

Step 1: Understanding potential energy.
The potential energy of an object is the energy stored due to its position or configuration. - In the case of a spring, potential energy increases when stretched and decreases when compressed. - Moving a body against gravity increases its potential energy, as work is done to lift the body.

Step 2: Rising air bubble.
When an air bubble rises in water, the pressure at the depth decreases. As the bubble rises, its potential energy decreases due to the reduction in depth.

Step 3: Conclusion.
Therefore, the potential energy decreases when an air bubble rises in water.

Step 1: Understanding the Concept:
Magnetic field at centre of a coil . When two coils are perpendicular, resultant field is vector sum.
Step 2: Detailed Explanation:
Let be field due to one coil. Since coils are perpendicular, fields are perpendicular to each other.
Resultant field = .
Ratio of field due to one coil to resultant = .
Step 3: Final Answer:
Thus, ratio = .

Step 1: Understanding the Concept:
Heat generated . Heat dissipated surface area . Also .
Step 2: Detailed Explanation:
At melting point: .
. So .
Step 3: Final Answer:
Thus, .

Step 1: Understanding the Concept:
Apparent weight at equator: . Given .
Step 2: Detailed Explanation:
.
.
rad s .
Step 3: Final Answer:
Thus, angular speed = rad s .

Step 1: Understanding the Concept:
Bernoulli's equation and continuity equation: , .
Step 2: Detailed Explanation:
m, m. , .
Continuity: .
Bernoulli: .
Given N m , .
.
m/s.
Rate of flow m /s ≈ .
Step 3: Final Answer:
Thus, rate of flow = m s .

Step 1: Understanding the Concept:
When suspension point accelerates upward, effective . Here m/s .
Step 2: Detailed Explanation:
, .
.
Step 3: Final Answer:
Thus, .

Step 1: Understanding the Concept:
Wave speed on string: . So .
Step 2: Detailed Explanation:
New speed .
.
Percentage increase = .
Step 3: Final Answer:
Thus, tension must increase by 44%.

Concept:

Step 1: Find potential energy


Step 2: Analyze shape

• Quadratic term upward curve initially
• Cubic term bends downward later So curve rises, then falls matches option (D) Conclusion:

Concept: Given:

Step 1: Tangential force


Step 2: Centripetal force


Step 3: Resultant force
Conclusion:

Step 1: Understanding the Concept:
.
Step 2: Detailed Explanation:


, not in options.
Step 3: Final Answer:
Angle is not among given options.

Step 1: Understanding the Concept:
Work done by variable force = change in kinetic energy.
Step 2: Detailed Explanation:


Integrate:



Step 3: Final Answer:
Final speed is 6 m/s.

Step 1: Understanding the Concept:
Power = Fv, need v in terms of S.
Step 2: Detailed Explanation:


Integrate:

Step 3: Final Answer:
Power is independent of displacement.

Concept: Gravitational potential energy at a distance from the center of Earth is: Thus, increase in potential energy when moving from to is: Also, .

Step 1: Identify initial and final positions.
Initial distance from Earth's center: Final distance:

Step 2: Apply the potential energy formula.


Step 3: Substitute .


Step 4: Express in terms of .
Since ,

Concept: Inside earth:

Step 1: Given condition.


Step 2: Solve.

Concept: Escape velocity:

Step 1: Kinetic energies.


Step 2: Ratio.

Concept: Momentum change in each bounce: After each bounce, momentum reduces by factor , forming a GP.

Step 1: Momentum series.


Step 2: Sum of GP.

Concept: From conservation of momentum: Recoil velocity is inversely proportional to rifle mass.

Step 1: Recoil velocity.


Step 2: Effect of mass.
Smaller mass (light rifle) larger recoil velocity.

Step 3: Injury relation.
Greater recoil velocity greater force on shoulder more injury.

Concept: Work done: If velocity remains constant, kinetic energy does not change.

Step 1: Find velocity.
Given: Velocity:

Step 2: Acceleration.
Since acceleration is constant but motion starts at , kinetic energy change in first 6 s is zero (no net external work effectively considered in given frame).

Step 3: Conclusion.

Concept: Work-energy theorem: Work done by opposing force is negative:

Step 1: Work done by force.


Step 2: Final kinetic energy.

Concept: Work-energy theorem for a system: This includes both kinetic and potential energy.

Step 1: Understanding forces.
All forces (internal + external) contribute to total work.

Step 2: Energy relation.


Step 3: Conclusion.
Thus, total work done changes total energy.

Concept: For heights comparable to Earth's radius, acceleration due to gravity is not constant. Use conservation of mechanical energy: Correct approach: where .

Step 1: Energy conservation equation. Cancel and simplify:

Step 2: Solve for .

Step 3: Substitute values.