a disc of mass 50 g slides with zero initial velocity down an inclined plane set at an angle 30 degrees to the horizontal. Having traversed a distance of 50 cm along the horizontal plane , the disc stops . Find the work done by the friction forces over the whole distance,assuming the friction coefficient 0.15 for both horizontal and inlined planes . take g = 10m/s^2

To find the work done by the friction forces on the disc as it slides down the inclined plane and then along the horizontal plane, we need to break the problem into parts and apply some physics principles. Let's go through the calculations step by step.

Step 1: Calculate the Forces Acting on the Disc

First, we need to determine the forces acting on the disc while it is on the inclined plane. The weight of the disc can be calculated using the formula:

Weight (W) = mass (m) × gravitational acceleration (g)

Given that the mass (m) is 50 g (which is 0.05 kg) and g is 10 m/s², we find:

W = 0.05 kg × 10 m/s² = 0.5 N

Components of Weight on the Incline

The weight can be resolved into two components: one parallel to the incline (which causes the disc to slide down) and one perpendicular to the incline (which affects the normal force).

The angle of the incline is 30 degrees, so:

• Parallel component (F_parallel) = W × sin(θ)
• Perpendicular component (F_{perpendicular}) = W × cos(θ)

Calculating these components:

F_parallel = 0.5 N × sin(30°) = 0.5 N × 0.5 = 0.25 N

F_{perpendicular} = 0.5 N × cos(30°) = 0.5 N × (√3/2) ≈ 0.433 N

Step 2: Calculate the Normal Force and Friction on the Incline

The normal force (N) on the incline is equal to the perpendicular component of the weight:

N = F_{perpendicular} ≈ 0.433 N

The friction force (F_friction) on the incline can be calculated using the coefficient of friction (μ = 0.15):

F_friction = μ × N

Thus:

F_friction = 0.15 × 0.433 N ≈ 0.065 N

Step 3: Calculate Work Done by Friction on the Incline

The work done by friction (W_friction) while the disc slides down the incline can be calculated using the formula:

W_friction = F_friction × d_incline

Assuming the distance along the incline is the same as the horizontal distance of 50 cm (0.5 m), we can calculate:

W_friction = 0.065 N × 0.5 m = 0.0325 J

Step 4: Analyze the Horizontal Motion

Once the disc reaches the horizontal plane, it continues to experience friction. The normal force on the horizontal plane is equal to the weight of the disc:

N_horizontal = W = 0.5 N

Calculating the friction force on the horizontal plane:

F_{friction, horizontal} = μ × N_horizontal = 0.15 × 0.5 N = 0.075 N

Step 5: Calculate Work Done by Friction on the Horizontal Plane

Now, we can find the work done by friction on the horizontal plane:

W_{friction, horizontal} = F_{friction, horizontal} × d_horizontal

Using the distance of 0.5 m:

W_{friction, horizontal} = 0.075 N × 0.5 m = 0.0375 J

Final Calculation: Total Work Done by Friction

The total work done by friction over the entire distance is the sum of the work done on the incline and the work done on the horizontal plane:

Total Work = W_friction + W_{friction, horizontal}

Total Work = 0.0325 J + 0.0375 J = 0.07 J

Therefore, the total work done by the friction forces over the whole distance is approximately 0.07 Joules.