Elastic potential energy Transitional Reasoning Practice

Questions

1. 1.

   A laboratory apparatus uses an ideal horizontal spring with a force constant of 500 N/m to propel a 0.25 kg slider along a track. The spring is compressed by 0.20 m from its equilibrium position and held against the slider, which is initially at rest. When released, the spring pushes the slider along a frictionless track until it separates at the equilibrium position and enters a rough horizontal surface where a constant friction force of 2.0 N opposes its motion. What distance does the slider travel across the rough surface before coming to rest?

   • A) 2.5 m
   • B) 5.0 m
   • C) 10.0 m
   • D) 20.0 m
2. 2.

   In an automated warehouse, an emergency deceleration mechanism uses an ideal horizontal spring with a force constant of 5000 N/m mounted against a rigid wall. A loaded robotic transfer cart of mass 200 kg travels along a level, frictionless guide rail at a constant speed of 10 m/s and engages the free end of the spring. What is the speed of the cart at the instant the spring has been compressed by 1.2 m from its equilibrium position?

   • A) 8.0 m/s
   • B) 6.0 m/s
   • C) 4.0 m/s
   • D) 2.0 m/s
3. 3.

   In a physics laboratory, an automated test cart attached to an ideal horizontal spring oscillates on a frictionless track. The cart is initially pulled to a distance $D$ from its equilibrium position and released from rest, storing a total mechanical energy composed entirely of elastic potential energy. As the cart moves toward the equilibrium position, it passes a sensor located at a displacement of $\frac{1}{2}D$ from equilibrium. What is the ratio of the cart's kinetic energy to the elastic potential energy stored in the spring, $\frac{K}{U}$, at the moment it passes the sensor?

   • A) $1$
   • B) $2$
   • C) $\frac{1}{3}$
   • D) $3$
4. 4.

   A student compresses an ideal vertical spring with a force constant of 600 N/m downward by 0.20 m from its equilibrium position to load a 0.40 kg projectile into a model launcher. When released from rest, the spring expands to its equilibrium position, transferring all of its stored elastic potential energy to the projectile. Neglecting air resistance and taking the acceleration due to gravity as 10 m/s², what is the maximum height reached by the projectile above its point of release?

   • A) 3.0 m
   • B) 1.5 m
   • C) 6.0 m
   • D) 12.0 m
5. 5.

   An industrial testing apparatus contains an ideal horizontal spring with a force constant of 400 N/m anchored to a rigid support. A technician initially compresses the spring by 0.30 m from its equilibrium position and then releases it, allowing it to expand past equilibrium until it is stretched by 0.10 m on the opposite side. What is the change in the elastic potential energy stored in the spring, $\Delta U = U_f - U_i$, between these two states?

   • A) +16.0 J
   • B) -32.0 J
   • C) -16.0 J
   • D) -20.0 J
6. 6.

   A safety engineer tests an emergency arrestor mechanism that incorporates an ideal linear spring. During a static calibration test, digital sensors record that the spring exerts a restoring force of magnitude 240 N while storing 36 J of elastic potential energy. Assuming the spring obeys Hooke's law, what is the compression of the spring from its equilibrium position at this instant?

   • A) 0.15 m
   • B) 0.45 m
   • C) 0.60 m
   • D) 0.30 m
7. 7.

   In a vehicle safety testing facility, an automated test cart of mass $m$ travels along a level, frictionless track at a constant speed $v$ and collides with a stationary bumper containing an ideal horizontal spring. The cart momentarily comes to rest when the bumper spring reaches its maximum compression, storing a peak elastic potential energy of $E_0$. In a subsequent trial, technicians increase the cart's mass to $2m$ and its speed to $2v$ before it collides with the exact same bumper. Assuming mechanical energy is conserved during the impact, what is the maximum elastic potential energy stored in the spring during the second trial?

   • A) $2E_0$
   • B) $4E_0$
   • C) $8E_0$
   • D) $16E_0$
8. 8.

   An engineering team tests two ideal linear springs, X and Y, for a tension-release mechanism. Spring X has a force constant of $k$, while spring Y has a force constant of $3k$. During a calibration test, identical pulling forces of magnitude $F$ are applied to both springs within their elastic limits, stretching them from their equilibrium lengths. If the elastic potential energy stored in spring X is $E$, what is the elastic potential energy stored in spring Y?

   • A) $9E$
   • B) $3E$
   • C) $\frac{1}{3}E$
   • D) $\frac{1}{9}E$
9. 9.

   A group of students investigates the energy storage of a shock absorber using an ideal linear spring in a mechanics lab. They find that compressing the spring from its equilibrium position by an initial distance of 0.10 m stores 4.0 J of elastic potential energy. Assuming the spring obeys Hooke's law, how much additional work must be performed on the spring to compress it by a further 0.10 m?

   • A) 8.0 J
   • B) 12.0 J
   • C) 16.0 J
   • D) 4.0 J
10. 10.

    A group of students designs a shock-absorbing bumper for a test cart using an ideal spring. In an initial calibration test, compressing the spring by a distance $x$ stores an elastic potential energy of 18 J. For a heavy-duty model, they replace the spring with a stiffer one having three times the original spring constant and compress it by a distance of $2x$. Assuming both springs obey Hooke's law, how much elastic potential energy is stored in the heavy-duty spring?

    • A) 216 J
    • B) 108 J
    • C) 72 J
    • D) 54 J

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