Acceleration Due To Gravity Of The Earth

Effect of Earth's Size and Mass on Gravity

If the Earth shrinks to half of its present size and its mass reduces to half of its actual mass, we need to determine the new acceleration due to gravity (g) on its surface.

Step 1: Acceleration due to Gravity Formula

The acceleration due to gravity (g) on the surface of a planet is given by:

Where:

• G is the gravitational constant.
• M is the mass of the planet.
• R is the radius of the planet.

Step 2: New Mass and Radius

The new mass (M′) is half of the original mass:

The new radius (R′) is half of the original radius:

Step 3: New Acceleration due to Gravity

The new acceleration due to gravity (g′) is:

Step 4: Relationship to Original g

Since , we can write:

Conclusion

If the Earth shrinks to half of its present size and its mass reduces to half of its actual mass, then the acceleration due to gravity (g) on its surface will be 2g.

The velocity vector of the particle is given as: In projectile motion:
- The horizontal component of velocity remains constant because no horizontal force acts on the object.
- The vertical component of velocity reverses its sign when the object returns to the same horizontal level (due to gravity).
Thus, at the time the body reaches the x-axis again:

Escape velocity ( ) is given by the formula: where is the acceleration due to gravity and is the radius of the planet.
For the given planet: Now, computing the escape velocity ratio: Thus, the ratio of escape velocity on the planet to that on Earth is 5:1.

In an isochoric process, the volume of the gas remains constant, which means that no work is done by the gas.
The first law of thermodynamics states:
where is the heat supplied, is the change in internal energy, and is the work done by the gas.
Since the volume is constant, . Therefore, the supplied heat is entirely used to change the internal energy of the gas, which in turn results in a change in the temperature of the gas, because internal energy is related to temperature in the case of an ideal gas.
Thus, in an isochoric process, the supplied heat increases both the internal energy and temperature of the gas.
Thus, the correct answer is option (D), internal energy and temperature.

We know the formula for current is: where is the charge and is the time. The current is given for a time of .
Now, the current is given by: So, the charge is: If the number of electrons is doubled, the total charge will also double. So the new charge will be: The new current is given by: where . Substituting the values: Thus, the current when twice the number of electrons flows for 20 seconds is .
Therefore, the correct answer is option (B), 3 A.

The total heat supplied in this process has two components:
1. The heat required to melt the ice at 0°C.
2. The heat required to increase the temperature of water from 0°C.
The heat required to melt the ice is given by the formula: where: - (mass of ice),
- (latent heat of fusion of ice).
So, the heat required to melt the ice is:
The total heat supplied is 418.72 J, and the heat required to melt the ice is 335 J.
Therefore, the remaining heat is used to increase the temperature of the water, which is: Thus, the quantity of heat used to increase the temperature of water from 0°C is 83.72 J, which corresponds to option (A).

We are given the following information: emf of the cell, current delivered, external resistance,
The total resistance in the circuit is the sum of the internal resistance and the external resistance : Using Ohm's law for the total circuit, we have: Substituting the given values: Solving for : Thus, the internal resistance of the cell is .
Therefore, the correct answer is option (B), 0.2 .

For capacitors connected in parallel, the total charge is the sum of the charges on each capacitor: where and are the charges on each capacitor.
The charge on each capacitor is related to the voltage across it by the formula: where is the capacitance and is the supply voltage.
Let the supply voltage be . Then, for each capacitor: The total charge is the sum of these charges: We are given that the total charge is . Thus: Since , we can simplify the equation: Thus, the supply voltage is 5 V.
Therefore, the correct answer is option (E), 5 V.

When two bodies at different temperatures are mixed, the final temperature of the mixture can be calculated using the principle of conservation of energy. The heat gained by the colder body is equal to the heat lost by the hotter body.
The equation for this can be written as: Where: - and are the masses of the two substances,
- and are their specific heats (for simplicity, assume both gases have similar specific heat values),
- and are their initial temperatures,
- is the final temperature.
Since we are given that both substances are in equal moles (0.5 mole each), and assuming similar specific heat values for and , we can simplify the equation by setting the specific heats and masses to equal values.
Thus, the equation simplifies to: Solving for : Thus, the final temperature of the mixture is .
Therefore, the correct answer is option (B), 34.5°C.

The energy required to excite a hydrogen atom between two energy levels is given by the difference in their energies.
The energy of the nth orbit of a hydrogen atom is given by: For the first excited state ( ) and the second excited state ( ): The energy required to excite the hydrogen atom from the first excited state to the second excited state is the difference: Thus, the energy required is 1.89 eV.