Answer:
1) Any particle moving in a horizontal plane slowed by friction, deceleration = 32 μ
2) The particle moving by acceleration = P/m - 32μ OR The external force = ma + 32μm
Explanation:
* Lets revise Newton’s Third Law:
- For every action there is a reaction, equal in magnitude and opposite
in direction.
- Examples:
# 1) A particle moving freely against friction in a horizontal plane
- When no external forces acts on the particle, then its equation of
motion is;
∵ ∑ forces in direction of motion = mass × acceleration
∵ No external force
∵ The friction force (F) = μR, where μ is coefficient of the frictional force
and R is the normal reaction of the weight of the particle on the
surface
∵ The frictional force is in opposite direction of the motion
∴ ∑ forces in the direction of motion = 0 - F
∴ 0 - F = mass × acceleration
- Substitute F by μR
∴ - μR = mass × acceleration
∵ R = mg where m is the mass of the particle and g is the acceleration
of gravity
∴ - μ(mg) = ma ⇒ a is the acceleration of motion
- By divide both sides by m
∴ - μ(g) = a
∵ The acceleration of gravity ≅ 32 feet/sec²
∴ a = - 32 μ
* Any particle moving in a horizontal plane slowed by friction,
deceleration = 32 μ
# 2) A particle moving under the action of an external force P in a
horizontal plane.
- When an external force P acts on the particle, then its equation
of motion is;
∵ ∑ forces in direction of motion = mass × acceleration
∵ The external force = P
∵ The friction force (F) = μR, where μ is coefficient of the frictional force
and R is the normal reaction of the weight of the particle on the
surface
∵ The frictional force is in opposite direction of the motion
∴ ∑ forces in the direction of motion = P - F
∴ P - F = mass × acceleration
- Substitute F by μR
∴ P - μR = mass × acceleration
∵ R = mg where m is the mass of the particle and g is the acceleration
of gravity
∴ P - μ(mg) = ma ⇒ a is the acceleration of motion
∵ The acceleration of gravity ≅ 32 feet/sec²
∴ P - 32μm = ma ⇒ (1)
- divide both side by m
∴ a = (P - 32μm)/m ⇒ divide the 2 terms in the bracket by m
∴ a = P/m - 32μ
* The particle moving by acceleration = P/m - 32μ
- If you want to fin the external force P use equation (1)
∵ P - 32μm = ma ⇒ add 32μm to both sides
∴ P = ma + 32μm
* The external force = ma + 32μm
the angle between the vector and the horizontal direction (x), the two sides are related to 
