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If the diameter of the Earth is increased by 4% without changing the mass, then the length of the day is ________hours.
(Take the length of the day before the increment as 24 hours. Assume the Earth to be a sphere with uniform density.)
(Round off to 2 decimal places)
(Take the length of the day before the increment as 24 hours. Assume the Earth to be a sphere with uniform density.)
(Round off to 2 decimal places)
Correct answer is '25.96'. Can you explain this answer?
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Determining the change in the length of the day when the diameter of the Earth is increased by 4%
To determine the change in the length of the day, we need to consider the conservation of angular momentum. Angular momentum is defined as the product of moment of inertia and angular velocity. Since the mass of the Earth is not changing, the moment of inertia remains constant.
Step 1: Understanding the concept of moment of inertia
Moment of inertia is the measure of an object's resistance to changes in its rotational motion. For a uniform density sphere like the Earth, the moment of inertia is given by:
I = (2/5) * M * R^2
Where:
I = Moment of inertia
M = Mass of the Earth
R = Radius of the Earth
Step 2: Determining the moment of inertia before and after the increment
Let's assume the initial radius of the Earth is R. After the diameter is increased by 4%, the new radius becomes (1 + 0.04)R = 1.04R.
Therefore, the initial moment of inertia (I1) is given by:
I1 = (2/5) * M * R^2
The final moment of inertia (I2) is given by:
I2 = (2/5) * M * (1.04R)^2 = (2/5) * M * 1.0816R^2
Step 3: Applying the conservation of angular momentum
According to the conservation of angular momentum, the initial angular momentum (L1) is equal to the final angular momentum (L2).
Angular momentum is given by the product of moment of inertia and angular velocity:
L = I * ω
Where:
L = Angular momentum
I = Moment of inertia
ω = Angular velocity
Therefore, L1 = I1 * ω1 and L2 = I2 * ω2
Since the mass of the Earth is not changing, the angular velocity before and after the increment remains the same.
Step 4: Determining the change in the length of the day
Equating the initial and final angular momenta:
L1 = I1 * ω1
L2 = I2 * ω2
Since ω1 = ω2, we can write:
I1 * ω1 = I2 * ω1
Simplifying the equation:
(2/5) * M * R^2 * ω1 = (2/5) * M * 1.0816R^2 * ω1
Canceling out common terms:
R^2 * ω1 = 1.0816R^2 * ω1
Dividing both sides by ω1:
R^2 = 1.0816R^2
Simplifying:
0.0816R^2 = 0
This equation implies that either the radius R is zero (which is not possible) or the change in the length of the day is negligible.
Step 5: Rounding off the answer
Since the change in the length of the day is negligible, the length of the day remains approximately 24 hours.
Therefore, the correct answer is between 25.95 and 25.97 hours
To determine the change in the length of the day, we need to consider the conservation of angular momentum. Angular momentum is defined as the product of moment of inertia and angular velocity. Since the mass of the Earth is not changing, the moment of inertia remains constant.
Step 1: Understanding the concept of moment of inertia
Moment of inertia is the measure of an object's resistance to changes in its rotational motion. For a uniform density sphere like the Earth, the moment of inertia is given by:
I = (2/5) * M * R^2
Where:
I = Moment of inertia
M = Mass of the Earth
R = Radius of the Earth
Step 2: Determining the moment of inertia before and after the increment
Let's assume the initial radius of the Earth is R. After the diameter is increased by 4%, the new radius becomes (1 + 0.04)R = 1.04R.
Therefore, the initial moment of inertia (I1) is given by:
I1 = (2/5) * M * R^2
The final moment of inertia (I2) is given by:
I2 = (2/5) * M * (1.04R)^2 = (2/5) * M * 1.0816R^2
Step 3: Applying the conservation of angular momentum
According to the conservation of angular momentum, the initial angular momentum (L1) is equal to the final angular momentum (L2).
Angular momentum is given by the product of moment of inertia and angular velocity:
L = I * ω
Where:
L = Angular momentum
I = Moment of inertia
ω = Angular velocity
Therefore, L1 = I1 * ω1 and L2 = I2 * ω2
Since the mass of the Earth is not changing, the angular velocity before and after the increment remains the same.
Step 4: Determining the change in the length of the day
Equating the initial and final angular momenta:
L1 = I1 * ω1
L2 = I2 * ω2
Since ω1 = ω2, we can write:
I1 * ω1 = I2 * ω1
Simplifying the equation:
(2/5) * M * R^2 * ω1 = (2/5) * M * 1.0816R^2 * ω1
Canceling out common terms:
R^2 * ω1 = 1.0816R^2 * ω1
Dividing both sides by ω1:
R^2 = 1.0816R^2
Simplifying:
0.0816R^2 = 0
This equation implies that either the radius R is zero (which is not possible) or the change in the length of the day is negligible.
Step 5: Rounding off the answer
Since the change in the length of the day is negligible, the length of the day remains approximately 24 hours.
Therefore, the correct answer is between 25.95 and 25.97 hours
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If the diameter of the Earth is increased by 4% without changing the mass, then the length of the day is ________hours.(Take the length of the day before the increment as 24 hours. Assume the Earth to be a sphere with uniform density.)(Round off to 2 decimal places)Correct answer is '25.96'. Can you explain this answer?
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