Conservative and nonconservative forces Transitional Reasoning Download PDF

Questions

1. 1.

   A robotics laboratory tests two distinct control forces acting on a mobile calibration unit traveling across a horizontal workspace. Force 1 is a central force directed toward a fixed origin whose magnitude depends only on the unit's distance from that origin, while Force 2 is a propulsion force that always acts tangent to the unit's instantaneous trajectory with a constant magnitude. Which statement correctly classifies these forces and provides the valid physical reasoning?

   • A) Force 1 is nonconservative because its orientation varies across the plane, whereas Force 2 is conservative because its magnitude remains constant regardless of position.
   • B) Both Force 1 and Force 2 are conservative because neither force explicitly depends on the elapsed time during the motion.
   • C) Force 1 is conservative because the work it performs depends only on the initial and final radial distances from the origin, whereas Force 2 is nonconservative because the work it performs depends on the actual path length traveled.
   • D) Force 1 is nonconservative because radial interactions prevent the definition of a potential energy function, whereas Force 2 is conservative because forces acting parallel to displacement always conserve mechanical energy.
2. 2.

   A research team investigates an unknown horizontal interaction force $F$ acting on a sensor probe moving along the x-axis. In Trial 1, the probe travels from $x = 0$ m to $x = 2$ m at a constant speed of 1 m/s, and the force does work $W_1$. In Trial 2, the probe travels across the same interval from $x = 0$ m to $x = 2$ m at a constant speed of 3 m/s, yielding work $W_2$. In Trial 3, the probe returns from $x = 2$ m to $x = 0$ m at 1 m/s, yielding work $W_3$. Which set of experimental outcomes conclusively demonstrates that force $F$ is conservative rather than a velocity-dependent nonconservative force?

   • A) $W_1 = W_2$ and $W_3 = -W_1$, because the work done by a conservative force is independent of speed and reverses sign when the path direction is reversed.
   • B) $W_2 > W_1$ and $W_3 = -W_1$, because a conservative force delivers more energy at higher speeds while still reversing sign on the return path.
   • C) $W_1 = W_2$ and $W_3 = W_1$, because a conservative force must perform work of the same sign regardless of the direction of displacement.
   • D) $W_2 < W_1$ and $W_3 = 0$, because higher speeds reduce the interaction time with the force, causing the return path to require zero work.
3. 3.

   A skydiver with a deployed parachute falls vertically downward at a constant terminal velocity. During this descent, the gravitational force does positive work while air resistance does negative work, resulting in zero net work on the skydiver. Which statement correctly explains the changes in the skydiver's kinetic energy and total mechanical energy during this motion?

   • A) The kinetic energy remains constant and the total mechanical energy is conserved, because zero net work means no energy can enter or leave the system.
   • B) The kinetic energy remains constant because the net work is zero, while the total mechanical energy decreases because the nonconservative drag force does negative work.
   • C) The kinetic energy remains constant because conservative forces do no work at constant velocity, while the total mechanical energy decreases because gravity reduces potential energy.
   • D) The kinetic energy remains constant because the drag force balances gravity, while the total mechanical energy increases because gravity does positive work on the descending skydiver.
4. 4.

   A laboratory carriage attached to an ideal horizontal spring oscillates along a straight track. In Configuration 1, track friction is negligible, and sensors show that the carriage has the exact same kinetic energy each time it passes the equilibrium position. In Configuration 2, a friction pad is engaged, and the carriage exhibits a progressively lower kinetic energy on each successive pass through equilibrium. Which statement correctly explains this difference based on the work done by conservative and nonconservative forces?

   • A) In Configuration 1, the spring force is conservative, meaning potential energy is uniquely determined by position and mechanical energy is conserved; in Configuration 2, kinetic friction is nonconservative, continually dissipating mechanical energy through path-dependent negative work.
   • B) In Configuration 1, the spring force is conservative because it does zero net work between the turning point and the center; in Configuration 2, kinetic friction is nonconservative because its work cancels to zero after every full oscillation.
   • C) In Configuration 1, mechanical energy is conserved because the spring force does not do any work on the carriage; in Configuration 2, kinetic friction is nonconservative because it converts kinetic energy into potential energy stored in the track.
   • D) In Configuration 1, the spring force is conservative because it always acts toward the equilibrium position; in Configuration 2, kinetic friction is nonconservative because the direction of the force is perpendicular to the carriage's displacement.
5. 5.

   Engineers test two different arresting barriers designed to stop a moving test vehicle of mass $m$ that enters each barrier with an initial kinetic energy $K$. Barrier 1 uses an ideal elastic spring mechanism (conservative force), whereas Barrier 2 uses a fluid-filled hydraulic dashpot (nonconservative viscous force). In both tests, the vehicle compresses the barrier along a straight horizontal track until it momentarily comes to rest at maximum displacement. Which statement correctly compares the energy transformations and the subsequent motion of the vehicle immediately after coming to rest?

   • A) Both barriers store the vehicle's initial kinetic energy as potential energy, causing the vehicle to rebound with its original speed in both tests because the total energy of each system is conserved.
   • B) Barrier 1 dissipates the vehicle's mechanical energy permanently into thermal energy, keeping the vehicle at rest, whereas Barrier 2 stores the energy as fluid pressure potential energy and forcefully ejects the vehicle outward.
   • C) Both barriers permanently remove all mechanical energy from the vehicle, so neither barrier exerts any force once the vehicle stops momentarily, leaving the vehicle at rest at maximum displacement in both tests.
   • D) Barrier 1 stores the kinetic energy as recoverable elastic potential energy that exerts a restoring force to propel the vehicle backward, whereas Barrier 2 converts mechanical energy into thermal energy and exerts zero force once the vehicle is stationary, leaving it at rest.
6. 6.

   A student asserts that a system's total mechanical energy is conserved only if no nonconservative forces act on any part of the system. To evaluate this claim, a lab group analyzes a simple pendulum swinging in a vacuum, where string tension acts on the bob, and a block sliding down a frictionless curved track, where the normal force acts on the block. Which statement correctly evaluates the student's assertion and explains the mechanical energy behavior in these systems?

   • A) The assertion is correct, because string tension and the normal force are conservative forces whose potential energy functions prevent any mechanical energy loss.
   • B) The assertion is correct, because nonconservative contact forces inevitably dissipate mechanical energy into internal thermal energy during motion.
   • C) The assertion is incorrect, because nonconservative forces always perform positive work that dynamically balances the negative work done by gravity.
   • D) The assertion is incorrect, because mechanical energy remains conserved whenever nonconservative forces do zero net work, which occurs here because both forces act perpendicular to the displacement.
7. 7.

   A group of students analyzes the motion of an electric rail cart on an incline track. When the cart coasts upward with the motor turned off and negligible friction, its total mechanical energy $E_{\text{mech}} = K + U$ remains constant as kinetic energy converts to gravitational potential energy. However, engaging the electric motor causes $E_{\text{mech}}$ to increase steadily, whereas applying the mechanical brakes causes $E_{\text{mech}}$ to decrease. Which statement correctly evaluates how conservative and nonconservative forces govern these mechanical energy changes?

   • A) Conservative forces can alter total mechanical energy in either direction, whereas nonconservative forces are strictly dissipative and can only reduce mechanical energy.
   • B) The motor force is conservative because it adds mechanical energy to the cart, while only dissipative forces such as friction are classified as nonconservative.
   • C) Conservative forces only transfer energy between kinetic and potential forms without altering total mechanical energy, whereas nonconservative forces alter total mechanical energy through positive or negative work.
   • D) Both gravity and the motor force are nonconservative because any force that changes an object's speed along an incline alters its mechanical energy.
8. 8.

   An engineering team is developing an automated energy-accounting program for a suspended test pod that moves between various workstations in a facility. While the software successfully models gravitational interactions using a potential energy function, it cannot assign a potential energy function to the drag force exerted by the surrounding air. Why is it physically impossible to define a potential energy function for air drag, and what is the consequence of its work on the pod's mechanical energy?

   • A) Because the work done by air drag depends on the specific path taken between two points, no unique potential energy value can be assigned to a position, and its negative work decreases the pod's mechanical energy.
   • B) Because the work done by air drag is independent of the path taken between two points, a potential energy function is mathematically undefined, and its work keeps the pod's mechanical energy constant.
   • C) Because air drag always acts perpendicular to the velocity vector, its work along any path is strictly zero, preventing any definition of potential energy or change in mechanical energy.
   • D) Because air drag transfers energy exclusively into gravitational fields, a separate potential energy function cannot be created, and its positive work increases the pod's mechanical energy.
9. 9.

   A testing cart on a track moves from point K to point L and then returns directly to point K along the same path under the influence of an attached ideal spring and kinetic friction. How does the total work done by the spring force compare to the total work done by the friction force for this complete round trip?

   • A) The spring force does net positive work because it restores the cart to its starting position, whereas friction does zero net work along a closed path.
   • B) Both the spring force and the friction force do zero net work because the cart's net displacement over the round trip is zero.
   • C) The spring force does zero net work because it is conservative, whereas friction does negative net work because it continuously opposes motion.
   • D) The spring force does negative net work due to energy dissipation, whereas friction does positive net work during the return journey.
10. 10.

    A laboratory cart is moved from a starting position to an elevated platform along two different paths: Track 1 is a steep, straight incline, whereas Track 2 is a longer, curved ramp. A student measures the work done by the gravitational force and then evaluates the effect of kinetic friction along both paths. Based on the fundamental definitions of conservative and nonconservative forces, which statement correctly evaluates the work done on the cart?

    • A) The work done by gravity is identical along both tracks because gravity is a conservative force, whereas the work done by kinetic friction depends on the path taken because friction is a nonconservative force.
    • B) The work done by gravity is greater along Track 2 due to the longer distance traversed, while friction performs identical work along both tracks because their endpoints are the same.
    • C) Both gravity and friction perform path-independent work because the overall displacement vector from the start to the target platform is identical for both routes.
    • D) The work done by gravity depends on the track shape because gravitational potential energy varies with slope, whereas friction always does zero net work over any path between two resting states.