A rat is running on ice with speed `v=pims^(-1)`. Suddenly he decides to turn by `90^(@)` and want of keep running with the same speed throughout. What is the least amount of time (in seocnds) he needs for such a turn? Suppose that rat's feet can move independently. Coefficieny of friction between rat's feet and ice is 0.125.
`("Given : "pi^(2)=g)`

Answer

To solve the problem of the rat turning 90 degrees while maintaining a constant speed on ice, we can follow these steps:

### Step 1: Understand the Forces Acting on the Rat
When the rat is turning, it experiences centripetal force that allows it to change direction. The frictional force between the rat's feet and the ice provides this centripetal force.

### Step 2: Set Up the Equations
The frictional force \( F_f \) can be expressed as:
\[ F_f = \mu \cdot N \]
where \( \mu \) is the coefficient of friction (0.125) and \( N \) is the normal force, which equals the weight of the rat \( mg \) (where \( g = \pi^2 \)). Thus:
\[ F_f = \mu \cdot mg \]

### Step 3: Relate the Centripetal Force to Friction
For circular motion, the centripetal force \( F_c \) required to keep the rat moving in a circle of radius \( r \) at speed \( v \) is given by:
\[ F_c = \frac{mv^2}{r} \]
Setting the frictional force equal to the centripetal force gives us:
\[ \mu mg = \frac{mv^2}{r} \]
We can cancel \( m \) from both sides:
\[ \mu g = \frac{v^2}{r} \]

### Step 4: Solve for the Radius \( r \)
Substituting \( g = \pi^2 \) and \( v = \pi \):
\[ 0.125 \cdot \pi^2 = \frac{\pi^2}{r} \]
Rearranging gives:
\[ r = \frac{\pi^2}{0.125 \cdot \pi^2} = \frac{1}{0.125} = 8 \text{ m} \]

### Step 5: Calculate the Distance for the Turn
The rat is making a quarter-circle turn, so the distance \( d \) traveled during the turn is:
\[ d = \frac{1}{4} \cdot 2\pi r = \frac{1}{4} \cdot 2\pi \cdot 8 = 4\pi \text{ m} \]

### Step 6: Calculate the Time Taken for the Turn
Using the formula for time \( t \):
\[ t = \frac{d}{v} \]
Substituting the values:
\[ t = \frac{4\pi}{\pi} = 4 \text{ seconds} \]

### Final Answer
The least amount of time the rat needs to turn 90 degrees while maintaining its speed is **4 seconds**.

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