Momentum

The correct answer is 6
By taking the system as man and trolley and using conservation of linear momentum :


= 6 m/s

Step 1: Formula for Angular Momentum

The angular momentum is given by:

Where:

• : Principal quantum number
• : Planck's constant

Step 2: Calculate for

Substitute :

Step 3: Calculate for

Substitute :

Step 4: Calculate Change in Angular Momentum ( )

The change in angular momentum is:

Conclusion:

The change in angular momentum is

Step 1: Relation between kinetic energy and momentum - Kinetic energy where is momentum and is mass.

Step 2: Calculate new kinetic energy - Initial momentum , kinetic energy New momentum New kinetic energy

Step 3: Calculate percentage increase in kinetic energy - Percentage increase is given by:

Final Answer: The percentage increase in kinetic energy is 125%.

The angular momentum of a satellite in orbit is given by: where is the mass of the satellite, is the orbital velocity, and is the radius of the orbit.
Step 1: Express the orbital velocity. The orbital velocity is: where is the gravitational constant, is the mass of the earth, and is the radius of the orbit. Substitute into the angular momentum equation:
Step 2: Analyze the new radius. The initial radius of the satellite is: where is the radius of the earth and is the height above the earth's surface. If the distance is increased by eight times its initial value, the new radius is:
Step 3: Calculate the new angular momentum. The new angular momentum is: Simplify: For the given problem, using approximate values:
Final Answer: The new angular momentum is:

To analyze the given statements, we need to examine the dimensions of the quantities mentioned.

1. Planck's constant and Angular momentum:
   • Planck's constant ( ): This is a fundamental constant with dimensions of action. Its dimensional formula is given by where:
     • [M] stands for Mass
     • [L] stands for Length
     • [T] stands for Time
   • Angular momentum ( ): The angular momentum of a particle is given by where is the radius vector, and is the linear momentum. Its dimensional formula is also .
   • Therefore, Statement I is true as Planck's constant and angular momentum have the same dimensions.
2. Linear momentum and Moment of force:
   • Linear momentum ( ): It is defined as the product of mass and velocity, and its dimensional formula is .
   • Moment of force (Torque, ): It is the product of force and the perpendicular distance from the pivot point. Its dimensional formula is .
   • It is clear that linear momentum and moment of force have different dimensions. Thus, Statement II is false.

In conclusion, the correct answer is: Statement I is true but Statement II is false.

The problem involves the calculation of angular momentum when an object reaches its maximum height. Let's solve the problem step-by-step:

1. When the object is projected at an angle with initial velocity , the components of the initial velocity are:

and

1. At maximum height, the vertical component of the velocity becomes zero, i.e., .
2. The horizontal component of the velocity remains constant throughout the motion:
3. With vertical velocity zero at maximum height, the only horizontal velocity at this point contributes to the angular momentum.
4. Time to reach maximum height is given by:

1. So, the horizontal distance covered (x-coordinate at maximum height) can be calculated as:

1. At maximum height, the position coordinates of the object are , since .
2. The angular momentum about the origin is given by the cross product of position vector and momentum:

1. Substituting, simplifies to:

1. The angular momentum is along the negative z-axis, as denoted by the negative sign in the above calculation.

Thus, the magnitude and direction of the angular momentum of the object with respect to the origin when it reaches the maximum height is along the negative z-axis.