A circular disc with a groove along its diameter is placed horizontally on a rough surface. A block of mass $1\,\text{kg}$ is placed as shown. The co-efficient of friction between the block and all surfaces of groove and horizontal surface in contact is $\mu = \frac{2}{5}$. The disc has an acceleration of $25\,\text{m/s}^{2}$ towards left. Find the acceleration of the block with respect to disc. Given $\cos \theta = \frac{4}{5}$, $\sin \theta = \frac{3}{5}$.

Step 1: Identify the given parameters.

The mass of the block, the coefficient of friction, the acceleration of the disc, and the trigonometric values are given.

$$m = 1\text{ kg}$$ $$\mu = \frac{2}{5} = 0.4$$ $$a = 25\text{ m/s}^2$$ $$\cos\theta = \frac{4}{5}$$ $$\sin\theta = \frac{3}{5}$$

Step 2: Calculate the pseudo force in the frame of the disc.

Since the disc accelerates to the left, a pseudo force acts on the block to the right.

$$F^* = ma$$ $$F^* = 1 \times 25 = 25\text{ N}$$

Step 3: Resolve the pseudo force along and perpendicular to the groove.

The components of the pseudo force relative to the groove are determined by the angle $\theta$.

$$F^*_{\parallel} = ma\cos\theta$$ $$F^*_{\parallel} = 25 \times \frac{4}{5} = 20\text{ N}$$ $$F^*_{\perp} = ma\sin\theta$$ $$F^*_{\perp} = 25 \times \frac{3}{5} = 15\text{ N}$$

Step 4: Determine the normal forces acting on the block.

The normal force from the wall of the groove balances the perpendicular component of the pseudo force.

$$N = F^*_{\perp} = 15\text{ N}$$

The normal force from the horizontal floor is equal to the weight of the block.

$$N_{\text{floor}} = mg$$ $$N_{\text{floor}} = 1 \times 10 = 10\text{ N}$$

Step 5: Calculate the friction forces.

Friction forces act on the block from both the vertical wall of the groove and the horizontal floor.

$$f_{\text{wall}} = \mu N$$ $$f_{\text{wall}} = 0.4 \times 15 = 6\text{ N}$$ $$f_{\text{floor}} = \mu mg$$ $$f_{\text{floor}} = 0.4 \times 10 = 4\text{ N}$$

Step 6: Verify if the block slides.

Compare the driving force along the groove with the maximum total friction force.

$$f_{\text{max}} = f_{\text{wall}} + f_{\text{floor}}$$ $$f_{\text{max}} = 6 + 4 = 10\text{ N}$$

Since $F^*_{\parallel} = 20\text{ N} > 10\text{ N}$, the block slides along the groove.

Step 7: Calculate the acceleration of the block relative to the disc.

Apply Newton's second law along the direction of the groove.

$$ma_{\text{rel}} = F^*_{\parallel} - f_{\text{floor}} - f_{\text{wall}}$$ $$ma_{\text{rel}} = 20 - 4 - 6 = 10\text{ N}$$ $$a_{\text{rel}} = \frac{10}{1} = 10\text{ m/s}^2$$

Answer: $10\text{ m/s}^2$ along the groove, toward the lower-right end.