Work And Energy

To find the power developed by a time-dependent force acting on a body, we follow these steps:

1. The force acting on the body is given by the vector function:

1. .
2. The velocity of the body, which is initially at rest, can be found by integrating the acceleration. The acceleration is given by:

1. .
2. Integrating the acceleration with respect to time to find the velocity:

1. .

• .

1. Now, compute the power developed by the force, which is given by:

1. (dot product of force and velocity vectors).
   • Substitute and :

• Simplifying gives:

1. Thus, the expression for the power developed by the force at any time is:

1. .

Therefore, the correct option is: (9t³ + 6t⁵) W.

To calculate the weight experienced by the man, we need to consider the component of the weight of the iron bar that acts along the line touching the man's shoulder, because the other component will act against the support provided by the ground.

The weight of the bar is given as 12 kg. When the rod makes an angle with the horizontal, the component of the weight acting along the direction of the man's shoulder can be found using the concept of resolving vectors.

The weight component along the vertical (which is relevant here as the force experienced by the bar due to gravity should be analyzed along its length) is given by:

Where:

• is the total weight of the rod (12 kg).
• is the angle with the horizontal ( ).

Let's substitute the values:

Calculating this gives:

However, since the weight experienced by the man is given in terms of mass under gravitational effect, convert Newtons back to mass equivalent in kg. Remember, 1 kg of force under Earth's gravity exerts approximately 9.8 N:

Revisiting the problem, this approach simplifies as:

The component of weight contributing to the man's experienced load will be horizontally resolved and should be:

which breaks to a result for visualization directly as 3 kg horizontally actually lifting the reference weight converting equivalent experience lifting.

Thus, the weight experienced by the man is 3 kg. The correct answer is the option .

Initial Potential Energy of the Ball:
When the ball is at height , its initial potential energy is:

where is the mass of the ball and is the acceleration due to gravity.

Potential Energy After Rebound:
After rebounding to a height of , the potential energy of the ball is:

Calculate the Percentage Loss of Energy:
The loss in energy is given by the difference between the initial and final potential energies:

The percentage loss in energy is:

Calculate the Velocity Before Striking the Ground:
Using energy conservation, the velocity of the ball just before it strikes the ground can be found from the initial potential energy:

Solving for :

Conclusion:
The percentage loss of total energy is 50%, and the velocity of the ball before it strikes the ground is .

Since the track is frictionless, we can use the principle of conservation of mechanical energy. At point A, the particle has potential energy and no kinetic energy, while at point B, it will have both kinetic and potential energy.

Calculate Potential Energy Difference Between Points A and B:

• The height of A is 1 m, and the height of B is 0.5 m.
• The difference in height, h, is .Apply Conservation of Mechanical Energy:

At point A, and . At point B, and .

Setting up the equation:

Simplify and solve for v:

To find the work done by the force as it displaces a body from m to m, we use the work integral:

First, compute the indefinite integral:

Now, apply the limits of integration from 2 to 4:

Calculate each term:

Substitute these into the equation:

The work done by the force is J, which fits within the provided range (58,58).

To solve the problem of finding the work done to stop a solid circular disc rolling on a floor, we need to consider both its translational and rotational kinetic energy. The total kinetic energy of the disc can be expressed as:

• Translational kinetic energy:
• Rotational kinetic energy:

where is the mass of the disc, is the speed of the center of mass, is the moment of inertia, and is the angular velocity.

For a solid disc, , and since the disc rolls without slipping, .

Substituting and , we find:

The total kinetic energy is the sum of translational and rotational components:

Substituting the known values:

Hence, the work done to stop the disc (which is equal to the reduction in total kinetic energy) is . This value matches the expected range .

To determine the ratio of the kinetic energy of the artillery to that of the shell, we can use the principles of conservation of momentum and the definition of kinetic energy.

1. Conservation of Momentum: where and are the velocities of the artillery and shell respectively after firing.
2. Kinetic Energy Formula: The kinetic energy of an object is given by: where is the mass and is the velocity of the object.
3. From the conservation of momentum, we can express the artillery's velocity in terms of the shell's velocity:
4. Kinetic Energy of Artillery:
5. Kinetic Energy of Shell:
6. Ratio of Kinetic Energies: The ratio of the kinetic energy of the artillery to that of the shell is:
7. Conclusion: Thus, the correct answer is , which is the ratio of the kinetic energy of the artillery to that of the shell.

This solution demonstrates the use of physics concepts such as conservation of momentum and kinetic energy, leading us to the correct answer.

To find the magnitude of the -component of force acting on the particle at point , we can use the concept of potential energy and force relationship. The force components are given by the negative gradient of the potential energy function.

The potential energy function is given as:

The force acting on the particle is given by:

Here, represents the gradient of the potential energy function . The gradient in three dimensions is:

Let's calculate each partial derivative:

Partial derivative with respect to :

Partial derivative with respect to :

Partial derivative with respect to :

Substituting these partial derivatives, the force is:

To find the -component of the force at , substitute , , and :

The magnitude of the -component of force is:

Thus, the magnitude of the -component of the force acting on the particle at point is 4 N.

The work done by a force on a particle moving through a displacement is given by the dot product: Given: The dot product is: Using the properties of the dot product: For the work to be zero: Thus, the value of is .

We are given:

• Mass
• Force

Step 1: Use Newton’s Second Law to find acceleration:

Step 2: Integrate to find velocity:

Step 3: Power is given by dot product :

Work done by the force — stepwise evaluation

1. Integral setup

The total work is the line integral split into the – and –parts:

2. Evaluate the –integral

3. Evaluate the –integral

4. Combine results

Final Answer:

To find the value of , we need to calculate the work done by the force as it displaces an object by 1 meter.

The work done by a force is given by the integral of the force over the displacement:

Given:

• The work done
• The displacement is from to

We now calculate the integral:

Substituting the limits of integration:

Equating it to the given work done:

Solve for :

• Subtract 1 from both sides:
  
• Multiply both sides by 3:
  N/m²

Therefore, the correct value of is 12 N/m².

This matches the correct answer from the given options.