A conveyor system pulls a 40\text{ kg} crate at a constant speed of 3\text{ m s}^{-1} up a… — JAMB UTME

A conveyor system pulls a $40\text{ kg}$ crate at a constant speed of $3\text{ m s}^{-1}$ up a rough inclined ramp. The ramp rises $3\text{ m}$ for every $5\text{ m}$ measured along its slope (giving $\sin\theta = 0.6$ and $\cos\theta = 0.8$ ). If the coefficient of kinetic friction between the crate and the ramp is $0.25$ and $g = 10\text{ m s}^{-2}$ , what is the power output of the conveyor system in watts?

Answer: 960 W

Answer

The power output required by the conveyor system is

960\text{ W}

To pull the crate up the ramp at constant speed, the conveyor force must overcome both the parallel gravitational component (

240\text{ N}

) and friction (

80\text{ N}

), making the total force

320\text{ N}

. Multiplying this force by the constant speed of

3\text{ m s}^{-1}

gives a total power output of

960\text{ W}

Step-by-Step Solution

Calculate the component of weight parallel to the inclined plane

F_g = mg \sin\theta = 40 \times 10 \times 0.6 = 240\text{ N}

Gravity pulls the object back down along the slope with force

mg \sin\theta

Calculate the normal reaction force perpendicular to the plane

N = mg \cos\theta = 40 \times 10 \times 0.8 = 320\text{ N}

The normal force balances the perpendicular weight component.

Determine the magnitude of kinetic friction force

f_k = \mu N = 0.25 \times 320 = 80\text{ N}

Friction opposes motion up the slope and depends on the normal force.

Calculate the total pulling force needed for zero net acceleration

F = F_g + f_k = 240 + 80 = 320\text{ N}

At constant velocity, net force along the incline must equal zero, so

F = mg\sin\theta + f_k

Calculate the power output of the conveyor

P = F \times v = 320\text{ N} \times 3\text{ m s}^{-1} = 960\text{ W}

Power developed by a constant force moving an object at velocity

v

is given by

P = Fv

Key Concept

Work done against gravity and friction, and rate of doing work (Power

P = Fv

)

Question

Answer

Step-by-Step Solution

Key Concept

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