Work Energy And Power

To solve the problem, we need to calculate the power developed by the engine as it drags a 5000 kg mass with a velocity of 5 m/s along a smooth inclined plane with an inclination of 1 in 50.

1. Understanding the Inclination:
A slope of "1 in 50" means that for every 50 meters along the incline, the vertical rise is 1 meter. Hence:

2. Force Along the Incline:
Since the plane is smooth (frictionless), the engine only needs to overcome the component of gravitational force along the incline:

Where:
,
(using for simpler calculation as per standard approximation),

3. Calculating Power:
Power is given by:

4. Converting to Kilowatts:

Final Answer:
The power of the engine is 5 kW.

Step 1: Understanding the Energy Transfer Formula In an elastic collision, the fraction of kinetic energy transferred from mass to mass is given by: Multiplying by 100 gives the percentage transfer.

Step 2: Substituting Given Values Given: Thus, the percentage of kinetic energy transferred is:

Step 1: Understanding the Principle of Moments The principle of moments states that for rotational equilibrium: Given: - A metre scale is balanced at its centre (50 cm) initially. - Two coins of mass each are placed at the 10 cm mark. - The new balance point shifts to 35 cm. - Let the mass of the metre scale be .

Step 2: Calculate the Moments The moment of the two coins about the new balance point: The moment of the metre scale's weight about the new balance point:

Step 3: Solve for Using the equilibrium condition: Thus, the mass of the metre scale is:

Step 1: Apply Newton’s Law of Cooling According to Newton’s Law of Cooling: where: - is the temperature of the body, - is the room temperature, - is a constant. Rearranging the equation: Taking two temperature conditions and solving for : where: - , - , - , - minutes, - minutes. Using the equation: Solving for :

Step 1: Understanding Energy Conservation in a Vertical Circle
- The total mechanical energy at any position is given by: - At the highest point, velocity is minimum and tension is zero.

Step 2: Finding Kinetic Energy at a Given Angle
- The maximum kinetic energy occurs at the lowest point. - At an angle , kinetic energy becomes half of this maximum value. - Using energy conservation: Solving for , we get:

Step 3: Conclusion
Since satisfies the condition, Option (1) is correct.

Step 1: The energy lost by the metal block will be used to melt the ice. The energy lost by the metal block is given by: where , , and .

Step 2: The energy required to melt the ice is: where is the latent heat of fusion and is the mass of the ice melted.

Step 3: Since all the energy from the metal is used to melt the ice:

Step 1: Compute the Displacement Vector The displacement vector is given by: Step 2: Compute the Work Done Work done is given by the dot product: Expanding the dot product: Thus, the work done in the process is:

Step 1: The energy stored in a stretched wire is given by: where: - is the energy stored, - is the tension in the wire, - is the length of the wire, - is the cross-sectional area of the wire, - is the elongation of the wire.

Step 2: The elongation is proportional to for a constant tension . Hence, we focus on the expression for the energy stored.

Step 3: For wires A and B, the energy stored is proportional to: Since the length of wire A is twice that of wire B ( ), and the diameter of wire A is twice that of wire B ( ), the cross-sectional area is proportional to , so:

Step 4: The ratio of the energies stored in wires A and B is:

Step 5: Thus, the ratio of the energies stored in wires A and B is , so Option (3) is correct.

Step 1: Apply Energy Conservation Using the conservation of mechanical energy: Since the body is rolling without slipping: The moment of inertia of a rolling object determines how much kinetic energy is converted into potential energy.

Step 2: Finding the Height Solving for : For a circular ring, , so: The problem states that , which matches the condition for a circular ring. Thus, the body is a circular ring.

For monatomic gases, the degrees of freedom are 3 (translational degrees of freedom). For diatomic gases, there are 5 degrees of freedom (3 translational + 2 rotational degrees of freedom). Thus, the ratio of the degrees of freedom for monatomic to diatomic gas molecules is: Therefore, the correct answer is 3:5 .

Step 1: The surface energy of a drop is given by: where is the radius of the drop, and is the surface tension.

Step 2: For the original drop, the radius is . The surface energy of the original drop is:

Step 3: After the drop breaks into 125 smaller drops, each drop has a radius . The surface energy of one small drop is:

Step 4: The total surface energy after the breakage is:

Step 5: The change in surface energy is: