a disc of mass 50g slides with zero initial velocity down an inclined plane set at an angle 30 degrees to the horizontal

having transversed a distance 50cm along the horizontal plane the disc stops

find the work performed by friction forces over the whole distance , assuming friction coefficient 0.15 for both inclined and horizontal planes

give solution

To determine the work done by friction forces on the disc as it slides down the inclined plane and then along the horizontal plane, we can break the problem into two parts: the motion down the incline and the motion along the horizontal surface. Let's go through the calculations step by step.

Step 1: Analyzing the Motion on the Inclined Plane

The first part of the journey is down the inclined plane. The mass of the disc is given as 50 grams, which we need to convert to kilograms for standard SI units:

• Mass (m) = 50 g = 0.05 kg

The angle of the incline is 30 degrees. The gravitational force acting on the disc can be broken down into two components: one parallel to the incline and one perpendicular to it. The force of gravity acting down the incline (F_gravity) is given by:

• F_gravity = m * g * sin(θ)

Where:

• g = 9.81 m/s² (acceleration due to gravity)
• θ = 30 degrees

Calculating this:

• F_gravity = 0.05 kg * 9.81 m/s² * sin(30°)
• F_gravity = 0.05 kg * 9.81 m/s² * 0.5 = 0.245 N

Step 2: Calculating the Normal Force

The normal force (N) acting on the disc is the component of the gravitational force acting perpendicular to the incline:

• N = m * g * cos(θ)

Calculating this gives:

• N = 0.05 kg * 9.81 m/s² * cos(30°)
• N = 0.05 kg * 9.81 m/s² * (√3/2) ≈ 0.424 N

Step 3: Finding the Frictional Force

The frictional force (F_friction) acting against the motion can be calculated using the coefficient of friction (μ = 0.15):

• F_friction = μ * N

Substituting the values:

• F_friction = 0.15 * 0.424 N ≈ 0.0636 N

Step 4: Work Done by Friction on the Incline

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

• W_friction = -F_friction * d_incline

Assuming the length of the incline (d_incline) can be calculated from the horizontal distance (50 cm) and the angle:

• d_incline = d_horizontal / cos(θ) = 0.5 m / cos(30°) ≈ 0.577 m

Now, substituting the values:

• W_friction = -0.0636 N * 0.577 m ≈ -0.0367 J

Step 5: Analyzing the Motion on the Horizontal Plane

Next, we consider the motion along the horizontal plane. The frictional force remains the same, as the coefficient of friction is constant. The distance traveled on the horizontal plane is 0.5 m.

Step 6: Work Done by Friction on the Horizontal Plane

Using the same formula for work done by friction:

• W_friction = -F_friction * d_horizontal

Substituting the values:

• W_friction = -0.0636 N * 0.5 m ≈ -0.0318 J

Final Calculation: Total Work Done by Friction

To find the total work done by friction over the entire distance, we simply add the work done on the incline and the work done on the horizontal plane:

• Total Work = W_{friction, incline} + W_{friction, horizontal}
• Total Work = -0.0367 J - 0.0318 J ≈ -0.0685 J

Thus, the total work performed by friction forces over the whole distance is approximately -0.0685 Joules. The negative sign indicates that the work done by friction opposes the motion of the disc.