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Difficulty: Very hardNewton's Second Law (F=ma)

A military recovery vehicle is using a winch to pull a disabled transport truck out of deep mud on a level surface. The disabled truck has a weight of 50,000 N50,000 \text{ N}. The mud exerts a constant resistive drag force of 12,000 N12,000 \text{ N} against the truck's motion. If the winch must accelerate the truck horizontally at a rate of 0.5 m/s20.5 \text{ m/s}^2, how much tension force must the winch cable provide? (Assume gravitational acceleration g=10 m/s2g = 10 \text{ m/s}^2)

  1. A
    22,000 N22,000 \text{ N}
  2. B
    37,000 N37,000 \text{ N}
  3. 14,500 N14,500 \text{ N}Answer
  4. D
    2,500 N2,500 \text{ N}

Answer

The winch cable must provide 14,500 N14,500 \text{ N} of tension force.
The correct answer is found by first converting the truck's weight (50,000 N50,000 \text{ N}) to its mass (5,000 kg5,000 \text{ kg}) using g=10 m/s2g = 10 \text{ m/s}^2. Next, the net force required to accelerate this mass at 0.5 m/s20.5 \text{ m/s}^2 is calculated as 2,500 N2,500 \text{ N} using F=maF = ma. Finally, because the mud exerts a 12,000 N12,000 \text{ N} opposing drag, the winch must pull with enough force to overcome the drag and still provide the 2,500 N2,500 \text{ N} net force. Therefore, the total required tension is 12,000 N+2,500 N=14,500 N12,000 \text{ N} + 2,500 \text{ N} = 14,500 \text{ N}.

Step-by-Step Solution

1
Determine the mass of the disabled truck from its given weight.
m=Wg=50,000 N10 m/s2=5,000 kgm = \frac{W}{g} = \frac{50,000 \text{ N}}{10 \text{ m/s}^2} = 5,000 \text{ kg}
Newton's second law (F=maF = ma) requires mass, not weight.
2
Calculate the required net force to achieve the target acceleration.
Fnet=ma=5,000 kg×0.5 m/s2=2,500 NF_{\text{net}} = ma = 5,000 \text{ kg} \times 0.5 \text{ m/s}^2 = 2,500 \text{ N}
This is the net force that must remain after all opposing forces are overcome.
3
Set up the force balance equation for the horizontal direction.
Fnet=Twinch−FdragF_{\text{net}} = T_{\text{winch}} - F_{\text{drag}}
The net force is the vector sum of the applied tension pulling forward and the drag force resisting backward.
4
Solve the equation for the required winch tension (TwinchT_{\text{winch}}).
2,500 N=Twinch−12,000 N  ⟹  Twinch=14,500 N2,500 \text{ N} = T_{\text{winch}} - 12,000 \text{ N} \implies T_{\text{winch}} = 14,500 \text{ N}
Adding the drag force to the required net force gives the total applied force.

Key Concept

Newton's Second Law applied to multi-force systems.
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Topics

Newton's Second Law (F=ma)

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