Work done by a variable force Transitional Reasoning Download PDF

Questions

1. 1.

   A robotic positioning carriage moves along a straight horizontal track aligned with the $x$-axis from $x = 0\text{ m}$ to $x = 4.0\text{ m}$. An automated actuator exerts a two-dimensional, position-dependent force on the carriage given by $\vec{F}(x) = (8x + 5)\hat{i} + (3x^2 - 4)\hat{j}$, where force is in newtons and $x$ is in meters. How much work is done by this force on the carriage over this displacement?

   • A) 84 J
   • B) 132 J
   • C) 64 J
   • D) 48 J
2. 2.

   In a tribology experiment testing surface coatings, a 10 kg test block is propelled horizontally along a straight track from $x = 0\text{ m}$ to $x = 3.0\text{ m}$. Due to a graded polymer coating, the coefficient of kinetic friction increases non-linearly with position according to $\mu_k(x) = 0.05 + 0.02x^2$, where $x$ is in meters. Taking the acceleration due to gravity as $g = 10\text{ m/s}^2$, what is the total work done by the friction force on the block over this displacement?

   • A) $-33\text{ J}$
   • B) $+33\text{ J}$
   • C) $-42\text{ J}$
   • D) $-69\text{ J}$
3. 3.

   A motorized tensioning rig pulls an offshore mooring line along a straight horizontal test track. The pulling force applied by the rig increases non-linearly with position according to $F(x) = 12x + 4x^3$, where $F$ is in newtons and $x$ is in meters. What is the total work done by this applied force on the line as its displacement increases from $x = 1.0\text{ m}$ to $x = 3.0\text{ m}$?

   • A) 112 J
   • B) 128 J
   • C) 135 J
   • D) 160 J
4. 4.

   During a high-speed track test, an autonomous vehicle travels along a straight horizontal path from $x = 1.0\text{ m}$ to $x = 3.0\text{ m}$. An experimental engine provides a forward thrust given by $F_{\text{engine}}(x) = 18x^2$, while a magnetic guide rail exerts an opposing resistive force given by $F_{\text{drag}}(x) = 6x$, with both forces in newtons and $x$ in meters. What is the net work done on the vehicle over this displacement?

   • A) 120 J
   • B) 135 J
   • C) 180 J
   • D) 132 J
5. 5.

   During a vertical shaft inspection, a winch pulls up a freely hanging, uniform cable of mass 20 kg and total length 6.0 m onto a platform at constant speed. Because the mass of the hanging segment steadily decreases as the cable is retrieved, the lifting force varies with the retrieved length $y$ according to $F(y) = \frac{m}{L}g(L - y)$. Taking $g = 10\text{ m/s}^2$, how much work does the winch do to reel in the entire cable?

   • A) 1200 J
   • B) 800 J
   • C) 600 J
   • D) 300 J
6. 6.

   An electromagnetic rail launcher applies a position-dependent horizontal driving force given by $F(x) = 75 - 3x^2$ on a test sled, where $F$ is in newtons and $x$ is in meters. The sled starts from rest at $x = 0\text{ m}$ and is propelled forward until the driving force drops to zero. How much work does the launcher perform on the sled during this propulsion phase?

   • A) 375 J
   • B) 250 J
   • C) 187.5 J
   • D) 125 J
7. 7.

   During a spacecraft docking simulation, an automated mechanism pushes a probe along a straight track against a repulsive magnetic force. The magnitude of this opposing force is given by $F(x) = \frac{72}{x^2}$, where $F$ is in newtons and $x$ is the separation distance in meters. How much work must the mechanism do on the probe to decrease the separation from $x = 6.0\text{ m}$ to $x = 2.0\text{ m}$ at constant speed?

   • A) 24 J
   • B) 40 J
   • C) 32 J
   • D) 72 J
8. 8.

   A deep-sea sampling probe uses a specialized non-linear elastic tether to deploy sensors smoothly. Calibration shows that the restoring force exerted by the tether depends on its stretch distance according to the relation $F(x) = 15\sqrt{x}$, where force is in newtons and $x$ is in meters. How much work must an external mechanism do to stretch the tether from an initial extension of $x = 1$ m to a final extension of $x = 4$ m?

   • A) 45 J
   • B) 67.5 J
   • C) 70 J
   • D) 80 J
9. 9.

   During a safety certification test, a hydraulic ram compresses an industrial buffer along a straight track. The resistive force increases linearly with displacement according to $F(x) = 500x$ from $x = 0$ m to $x = 0.2$ m, where force is in newtons and displacement is in meters, reaching a maximum force of 100 N. For the remainder of the compression from $x = 0.2$ m to $x = 0.6$ m, the buffer yields and maintains a constant resistive force of 100 N. What is the total work done by the hydraulic ram to compress the buffer from $x = 0$ m to $x = 0.6$ m?

   • A) 30 J
   • B) 40 J
   • C) 50 J
   • D) 60 J
10. 10.

    A prototype maglev carrier moves along a straight horizontal rail where an automated propulsion system exerts a position-dependent force parallel to displacement given by $F(x) = (12x - 3x^2)$ N, with position $x$ measured in meters. How much work is done by this variable force on the carrier as it travels from $x = 1$ m to $x = 3$ m?

    • A) 18 J
    • B) 22 J
    • C) 27 J
    • D) 32 J