Introduction:
When a body slides down an inclined plane, the force of gravity acts vertically downwards while the normal force acts perpendicular to the plane. Additionally, there is a frictional force opposing the motion of the body. In this scenario, we are given the angle of inclination, coefficient of friction, and acceleration due to gravity. We need to calculate the acceleration of the body downwards.

Given:
Angle of inclination (θ) = 45°
Coefficient of friction (μ) = 0.5
Acceleration due to gravity (g) = 9.8 m/s²

Understanding the Forces:
1. Force of Gravity (Fg): This force acts vertically downward and is given by the equation Fg = mg, where m is the mass of the body.
2. Normal Force (Fn): This force acts perpendicular to the inclined plane and is equal in magnitude but opposite in direction to the component of the force of gravity acting perpendicular to the plane.
3. Frictional Force (Ff): This force opposes the motion of the body and is given by the equation Ff = μFn, where μ is the coefficient of friction.

Calculating the Forces:
1. Force of Gravity: As the body is sliding down the inclined plane, the force of gravity acts vertically downward and can be resolved into two components:
- Parallel to the inclined plane: Fg_parallel = mg sin(θ)
- Perpendicular to the inclined plane: Fg_perpendicular = mg cos(θ)
2. Normal Force: The normal force acts perpendicular to the inclined plane and is equal in magnitude but opposite in direction to the component of the force of gravity acting perpendicular to the plane. Therefore, Fn = Fg_perpendicular = mg cos(θ)
3. Frictional Force: The frictional force opposes the motion of the body and is given by the equation Ff = μFn = μmg cos(θ)

Calculating the Acceleration:
The net force acting on the body can be calculated by subtracting the frictional force from the parallel component of the force of gravity:
Net Force (Fnet) = Fg_parallel - Ff = mg sin(θ) - μmg cos(θ)

Using Newton's second law (F = ma), we can equate the net force to the product of mass and acceleration:
Fnet = ma

Substituting the values, we get:
mg sin(θ) - μmg cos(θ) = ma

Simplifying the equation, we have:
g sin(θ) - μg cos(θ) = a

Substituting the given values, we get:
9.8 * sin(45°) - 0.5 * 9.8 * cos(45°) = a

Solving the above equation, we find:
a ≈ 6.862 m/s²

Therefore, the acceleration of the body downwards is approximately 6.862 m/s².

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