An inverted ice-cream cone is kept fixed on a frictionless surface with its base lying on the surface. A block kept on its slanted surface slides down with constant velocity, when the cone is cut along the along the slant surface of the cone and laid flat on the surface, it forms a semicircle the coefficient of friction on the surface of the cone is `mu`. then find the value of

Explanation

To solve the problem step by step, we will analyze the forces acting on the block on the slanted surface of the cone and derive the expression for the coefficient of friction, \( \mu \).

### Step 1: Understand the Geometry of the Cone
- We have an inverted ice-cream cone with a base radius \( r \) and height \( h \).
- The block is sliding down the slanted surface of the cone at a constant velocity, which indicates that the net force acting on it is zero.

### Step 2: Identify Forces Acting on the Block
- The forces acting on the block are:
- Gravitational force, \( mg \) (acting downwards).
- Normal force, \( N \) (acting perpendicular to the surface of the cone).
- Frictional force, \( f \) (acting up the slope, opposing the motion).

### Step 3: Resolve the Gravitational Force
- The gravitational force can be resolved into two components:
- Parallel to the surface of the cone: \( mg \sin \theta \)
- Perpendicular to the surface of the cone: \( mg \cos \theta \)

### Step 4: Apply Newton's Second Law
- Since the block slides down with constant velocity, the net force along the slope is zero:
\[ mg \sin \theta = f \]
- The frictional force can be expressed as:
\[ f = \mu N \]
- The normal force \( N \) is equal to the perpendicular component of the gravitational force:
\[ N = mg \cos \theta \]

### Step 5: Set Up the Equation
- Substituting the expression for \( N \) into the frictional force equation:
\[ mg \sin \theta = \mu (mg \cos \theta) \]
- Dividing both sides by \( mg \) (assuming \( m \neq 0 \)):
\[ \sin \theta = \mu \cos \theta \]
- Rearranging gives:
\[ \tan \theta = \mu \]

### Step 6: Relate the Cone's Geometry to the Angle
- When the cone is cut along the slant surface and laid flat, it forms a semicircle with a radius \( R \).
- The relationship between the radius of the cone and the height can be expressed as:
\[ r = \frac{R}{2} \]
- The angle \( \theta \) can be determined from the geometry of the cone.

### Step 7: Find the Angle \( \theta \)
- From the geometry of the cone, we can find \( \cos \theta \):
\[ \cos \theta = \frac{r}{h} \]
- Since \( r = \frac{R}{2} \) and \( h = R \):
\[ \cos \theta = \frac{\frac{R}{2}}{R} = \frac{1}{2} \]
- Therefore, \( \theta = 60^\circ \).

### Step 8: Calculate \( \mu \)
- Using the relationship \( \tan \theta = \mu \):
\[ \tan 60^\circ = \sqrt{3} \]
- Thus, \( \mu = \sqrt{3} \).

### Step 9: Find \( \mu^2 \)
- Finally, we need to find \( \mu^2 \):
\[ \mu^2 = (\sqrt{3})^2 = 3 \]

### Final Answer
The value of \( \mu^2 \) is \( 3 \).

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