If the earth were to spin faster, acceleration due to gravity at the poles :

Options

A. Increases
B. Decreases
C. Remains the same
D. depends on how fast it spins

Answer

To solve the question regarding the effect of the Earth's increased spin on the acceleration due to gravity at the poles, we can follow these steps:

### Step 1: Understand the Forces at Play
When an object is on the surface of the Earth, it experiences two main forces: the gravitational force pulling it downwards (mg) and the centrifugal force due to the Earth's rotation. The centrifugal force acts outward and is dependent on the angular velocity (ω) of the Earth's rotation.

### Step 2: Write the Effective Gravitational Force Equation
The effective gravitational force (G_eff) experienced by an object at latitude θ can be expressed as:
\[ G_{\text{eff}} = g - R \omega^2 \cos^2(\theta) \]
Where:
- \( g \) is the acceleration due to gravity at the surface of the Earth.
- \( R \) is the radius of the Earth.
- \( \omega \) is the angular velocity of the Earth's rotation.
- \( \theta \) is the latitude.

### Step 3: Analyze the Situation at the Poles
At the poles, the latitude θ is 90 degrees. Therefore, we can substitute θ into the equation:
\[ \cos(90^\circ) = 0 \]
This means:
\[ G_{\text{eff}} = g - R \omega^2 \cdot 0 \]
Thus:
\[ G_{\text{eff}} = g \]

### Step 4: Conclusion
Since the centrifugal force becomes zero at the poles (due to the cosine term being zero), the acceleration due to gravity at the poles remains unaffected by the angular velocity of the Earth. Therefore, even if the Earth were to spin faster, the acceleration due to gravity at the poles would still be equal to \( g \).

### Final Answer
The acceleration due to gravity at the poles does not depend on the spin of the Earth, and it remains equal to \( g \).

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