**2 Work-energy principle Nonconservative force Motion on inclined plane with friction – Problems and Solutions**

1. A block slides down an inclined plane with friction. What is the block’s velocity when the block hits the ground? The coefficient of kinetic friction is 0.4. Acceleration due to gravity is 10 m/s^{2}.

__Known :__

Initial height (h_{o}) = 6 m

Final height (h_{t}) = 0 m

Initial speed (v_{o}) = 0 *(block initially at rest)*

Coefficient of kinetic friction (μ_{k}) = 0.4

Acceleration due to gravity (g) = 10 m.s^{-2}

cos θ = adj/hyp = 8/10

The vertical component of weight = w_{y} = w cos θ = m g cos θ = m (10)(8/10) = m (10)(4/5) = m (40/5) = 8 m

The normal force = N = w_{y} = 8 m

The force of kinetic friction = f_{k }= μ_{k} N = μ_{k} w_{y }= (0.4)(8 m) = 3.2 m

__Wanted : __Final speed (v_{t})

__Solution :__

The work-mechanical energy principle states that the work done by the nonconservative forces acting on an object is equal to the total change in kinetic and potential energies.

W_{nc} = ΔEM

W_{nc} = ΔEK + ΔEP

*W*_{nc}* = Work done by nonconservative force, **Δ**EM = The change in mechanical energy, **Δ**EK = The change in kinetic energy, **Δ**EP = The change in potential energy.*

The change in kinetic energy :

ΔEK = 1/2 m (v_{t}^{2} – v_{o}^{2}) = 1/2 m (v_{t}^{2} – 0) = 1/2 m v_{t}^{2}

The change in potential energy :

ΔEP = m g (h_{t} – h_{o}) = m (10)(0-6) = m (10)(-6) = – 60 m

Work done by the force of kinetic friction :

W_{nc} = – f_{k} s = – (3.2 m)(10) = – 32 m

*Minus sign indicates that the work done by the force of kinetic friction on the block is negative.*

Determine the final speed (v_{t}) :

W_{nc} = ΔEK + ΔEP

– 32 m = 1/2 m v_{t}^{2 } – 60 m

– 32 m = m (1/2 v_{t}^{2 } – 60)

– 32 = 1/2 v_{t}^{2 } – 60

– 32 + 60 = 1/2 v_{t}^{2}

28 = 1/2 v_{t}^{2}

2 (28) = v_{t}^{2}

56 = v_{t}^{2}

v_{t} = √4.14

v_{t} = 2√14 m.s^{-1}

2.

A block slides down on an inclined plane with friction. The final speed of the block when it hits the ground is 10 m/s^{2}. If the friction force is 2 N and acceleration due to gravity is 10 m/s^{2}. What is the height of h?

__Known :__

Mass of block (m) = 1 kg

Initial speed (v_{o}) = 0 *(block initially at rest)*

Final speed (v_{t}) = 10 ms^{-1}

Initial height (h_{o}) = h

Final height (h_{t}) = 0

Force of kinetic friction (f_{k}) = 2 N

Acceleration due to gravity (g) = 10 ms^{-2}

__Wanted :__ Height (h)

__Solution :__

The work is done by the force of kinetic friction :

W_{nc} = – f_{k} d = – (2)(15) = – 30

*Minus sign indicates that the work done by the force of kinetic friction on the block is negative.*

The change in kinetic energy :

ΔEK = 1/2 m (v_{t}^{2} – v_{o}^{2}) = 1/2 (1)(10^{2} – 0) = 1/2 (10^{2}) = 1/2 (100) = 50

The change in potential energy :

ΔEP = m g (h_{t} – h_{o}) = (1)(10)(0-h) = (10)(-h) = -10 h

The equation of the work-mechanical energy principle :

W_{nc} = ΔEK + ΔEP

– 30 = 50 – 10 h

10 h = 50 + 30

10 h = 80

h = 80/10

h = 8 meters

**What is the work-energy principle?****Answer:**The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. Mathematically, $W=ΔKE$, where $W$ is the work done and $ΔKE$ is the change in kinetic energy.

**How does a nonconservative force differ from a conservative force?****Answer:**A nonconservative force, like friction, results in a loss of mechanical energy from a system, typically in the form of heat. In contrast, conservative forces, like gravity or a spring force, don’t dissipate mechanical energy but rather can convert it between potential and kinetic energy within the system.

**How does friction affect the work done on an object moving on an inclined plane?****Answer:**Friction acts in a direction opposite to the motion of the object. This means that friction does negative work on the object, reducing its kinetic energy or increasing the amount of external work required to move it up the incline.

**Why doesn’t an object’s potential energy change due to nonconservative forces?****Answer:**Potential energy is associated with conservative forces, and its change depends on initial and final positions only, not on the path taken. Nonconservative forces can change the kinetic energy of an object but do not have associated potential energy.

**How does the angle of inclination influence the motion of an object on an inclined plane with friction?****Answer:**The greater the angle of inclination, the larger the component of gravitational force acting parallel to the plane. This makes the object harder to keep at rest or move up the plane, and easier to accelerate down the plane. Friction acts to resist this motion, but its effect becomes relatively less significant as the angle increases.

**Why do objects not accelerate indefinitely on an inclined plane with friction?****Answer:**Due to the presence of friction, as the object moves, some of the gravitational potential energy is transformed into heat. Eventually, the force due to gravity parallel to the plane will be balanced out by the frictional force, and the object will move with a constant velocity or come to rest.

**Can the work done by a nonconservative force be recovered?****Answer:**The work done by nonconservative forces usually gets transformed into forms of energy like heat, which are generally more challenging to convert back into useful mechanical energy. In most practical scenarios, this energy is considered “lost” to the system.

**What determines the magnitude of the frictional force on an inclined plane?****Answer:**The frictional force on an inclined plane is determined by the normal (perpendicular) force and the coefficient of friction between the surfaces. It’s given by $f_{friction}=μ×f_{normal}$, where $μ$ is the coefficient of friction.

**Why is potential energy defined as zero at some reference point?****Answer:**Potential energy is a relative measure. By defining a reference point where potential energy is zero, it provides a consistent basis for calculating changes in potential energy for various positions relative to that reference.

**How would the motion of an object on an inclined plane change if there were no friction?**

**Answer:**Without friction, the only force acting on the object along the plane would be the component of gravity parallel to the incline. As a result, the object would accelerate down the plane solely due to this force and not reach a constant velocity unless acted upon by some other external force.