Gravitational potential energy Transitional Reasoning Practice

Questions

1. 1.

   A testing apparatus consists of a lightweight rod of length 2.5 m with a concentrated 6.0 kg sensor module attached to its free end, pivoted from a fixed support. The sensor is held at an angle $\theta$ with the downward vertical such that $\cos\theta = 0.80$. Taking the lowest point of its swing as the reference level where gravitational potential energy is zero and $g = 9.8\text{ m/s}^2$, what is the gravitational potential energy of the sensor-Earth system at this position?

   • A) 117.6 J
   • B) 88.2 J
   • C) 147.0 J
   • D) 29.4 J
2. 2.

   A pumped-storage hydroelectric plant transfers $1.0 \times 10^4\text{ kg}$ of water from a lower reservoir at an elevation of 180 m above sea level to an upper reservoir at an elevation of 430 m above sea level. During the transfer, the electric pumps consume a total of 30.0 MJ of energy to overcome pipe friction and elevate the water. Taking the acceleration due to gravity as $g = 9.8\text{ m/s}^2$, what is the increase in the gravitational potential energy of the water-Earth system, and how much energy is dissipated as thermal energy?

   • A) The gain in gravitational potential energy is 30.0 MJ, and the energy dissipated as thermal energy is 0.0 MJ.
   • B) The gain in gravitational potential energy is 17.6 MJ, and the energy dissipated as thermal energy is 12.4 MJ.
   • C) The gain in gravitational potential energy is 24.5 MJ, and the energy dissipated as thermal energy is 5.5 MJ.
   • D) The gain in gravitational potential energy is 42.1 MJ, and the energy dissipated as thermal energy is 12.1 MJ.
3. 3.

   An aerospace engineering team tests a drop-mechanism component on Earth, where the local gravitational field strength is $g$. Lifting a package of mass $m$ by a vertical height $h$ above a reference datum increases the gravitational potential energy of the package-Earth system by $\Delta E_p$. For an upcoming lunar surface mission where the local gravitational field strength is $\frac{1}{6}g$, the team uses a modified test package of mass $2m$. By what factor must the vertical lifting height be adjusted on the Moon so that the package-Moon system gains the exact same gravitational potential energy $\Delta E_p$?

   • A) The vertical height must be $\frac{1}{3}h$.
   • B) The vertical height must be $\frac{1}{2}h$.
   • C) The vertical height must be $6h$.
   • D) The vertical height must be $3h$.
4. 4.

   A flight-testing team analyzes telemetry data from an automated research drone ascending vertically in a uniform gravitational field. During the first stage, the drone climbs through a vertical height of 20 m while accelerating from rest to a cruising speed. In the second stage, it climbs through the next 20 m at a constant vertical speed, and in the third stage, it climbs an additional 20 m while decelerating to a momentary hover. How does the increase in the gravitational potential energy of the drone-Earth system compare across these three successive 20 m stages?

   • A) The increase in gravitational potential energy is identical across all three stages because the vertical displacement is the same in each stage.
   • B) The increase in gravitational potential energy is greatest in the first stage because the drone undergoes a positive upward acceleration.
   • C) The increase in gravitational potential energy is greatest in the second stage because the drone travels at its highest constant velocity throughout that interval.
   • D) The increase in gravitational potential energy is smallest in the third stage because the upward deceleration reduces the energy stored in the field.
5. 5.

   An atmospheric research probe with a mass of 50 kg is released from an aircraft at an altitude of 1200 m and falls vertically to an altitude of 400 m, where its parachute deploys. Throughout this descent, atmospheric drag acts continuously on the falling probe. Taking the acceleration due to gravity as 9.8 m/s², what is the change in the gravitational potential energy of the probe-Earth system, and what is the work done on the probe by the gravitational force during this 800 m descent?

   • A) The gravitational potential energy decreases by 392 kJ, and the work done by gravity is -392 kJ.
   • B) The gravitational potential energy increases by 392 kJ, and the work done by gravity is +392 kJ.
   • C) The gravitational potential energy decreases by 392 kJ, and the work done by gravity is +392 kJ.
   • D) The gravitational potential energy decreases by an amount less than 392 kJ due to atmospheric drag, and the work done by gravity is +392 kJ.
6. 6.

   A freight elevator system uses a motor and an overhead cable connected over a frictionless pulley to link an elevator car to a heavy counterweight. During a maintenance test, the empty car with a mass of 600 kg descends vertically by 15 m at a constant speed, causing the 800 kg counterweight to rise vertically by 15 m. Taking the acceleration due to gravity as 9.8 m/s², what is the net change in the gravitational potential energy of the car-counterweight system?

   • A) It decreases by 29.4 kJ.
   • B) It increases by 29.4 kJ.
   • C) It increases by 205.8 kJ.
   • D) It remains unchanged because both masses move through the same vertical distance.
7. 7.

   A construction team uses a pivot crane to raise a uniform steel girder of mass 50 kg and length 4.0 m from lying flat on the ground to standing vertically upright on one end. The ground is chosen as the reference level where gravitational potential energy is zero, and the girder's cross-sectional thickness is negligible. Taking g = 9.8 m/s2, what is the increase in the gravitational potential energy of the girder?

   • A) 1960 J
   • B) 980 J
   • C) 490 J
   • D) 3920 J
8. 8.

   During a laboratory investigation, three students move identical 2.0 kg carts from a flat laboratory floor to a benchtop located at a vertical height of 1.2 m. Student 1 lifts Cart 1 vertically straight up to the bench at a constant speed. Student 2 pushes Cart 2 at a constant speed along a smooth, frictionless inclined ramp that is 2.4 m long. Student 3 moves Cart 3 along a stepped track containing alternating horizontal and vertical segments. Taking the floor as the zero reference level for gravitational potential energy, which statement correctly compares the gravitational potential energy gained by the carts?

   • A) Cart 2 gains the greatest gravitational potential energy because it travels the greatest total distance along the inclined ramp.
   • B) Cart 1 gains the greatest gravitational potential energy because its motion is oriented entirely opposite to the direction of gravity.
   • C) All three carts gain the same gravitational potential energy because the change depends only on vertical displacement, regardless of the path taken.
   • D) Cart 3 gains less gravitational potential energy than Cart 1 because moving along horizontal segments does not store energy.
9. 9.

   A maintenance crew uses a hoist to lift a 250 kg equipment crate from an underground basement located 6.0 m below ground level to a rooftop platform located 14.0 m above ground level. An apprentice calculates the gravitational potential energy of the crate using ground level as the zero reference datum, while a senior engineer sets the zero reference datum at the basement floor. Taking the acceleration due to gravity as $g = 9.8\text{ m/s}^2$, which statement correctly compares their calculated values for the crate's gravitational potential energy at the rooftop and the net change in its gravitational potential energy ($\Delta E_p$) between the two levels?

   • A) Both observers calculate identical values for the rooftop gravitational potential energy and identical values for $\Delta E_p$.
   • B) The senior engineer calculates a smaller rooftop gravitational potential energy than the apprentice, but both calculate identical values for $\Delta E_p$.
   • C) The senior engineer calculates a greater rooftop gravitational potential energy than the apprentice, but both calculate identical values for $\Delta E_p$.
   • D) The apprentice calculates a greater rooftop gravitational potential energy and a larger $\Delta E_p$ than the senior engineer.
10. 10.

    In an automated distribution center, a robotic lift moves two crates from the floor to different storage racks. Crate X has a mass of 12 kg and is placed at a vertical height of 2.0 m, while Crate Y has a mass of 8.0 kg and is placed at a vertical height of 4.5 m. Taking the floor as the reference level where gravitational potential energy is zero, how does the gravitational potential energy gained by Crate Y compare to that gained by Crate X?

    • A) Crate X gains more gravitational potential energy than Crate Y because potential energy is primarily determined by mass.
    • B) Crate Y gains 1.5 times the gravitational potential energy of Crate X because its greater elevation compensates for its lower mass.
    • C) Both crates gain the same gravitational potential energy because the ratio of their masses exactly balances the ratio of their heights.
    • D) Crate Y gains 2.25 times the gravitational potential energy of Crate X because gravitational potential energy is proportional to the square of the height.

Answers and step-by-step solutions appear after you finish the test.