Intro to Conservation of Energy Practice Problems

Draw the wave function corresponding to n=5 for the potential energy depicted in the figure below.

A driver starts his car resting on top of a semicircular hill. The road leading to the foothills is frictionless and curved. Derive an equation to calculate the car's speed at an angle θ as depicted in the figure below.

A 650 g bird is stuck on the upper free end of a vertical spring with a spring constant of 14 N/m. The bottom of the ideal spring is tied to the earth. The bird trying to liberate itself from the spring flies vertically at a time t = 0 s, producing a constant vertical force of 12 N. The spring has been stretched 64 cm at time t. Determine the speed of the bird at time t.

A physics student built a rotating device using a 1.1 m bar of negligible mass and a 75-gram sphere fixed to one end of the bar. The opposite end is supported by a fixed frictionless horizontal axis, making it possible to freely rotate the bar in a vertical plane. Find the minimum speed that the student should give to the sphere at its lowest position so that the sphere makes a full rotation.

A toy designer creates a roller coaster-type track using a smooth ramp and a frictionless vertical loop of diameter d. The car starting at rest slides down the incline and should go around the vertical loop by always being in contact with the loop. The designer asks a physicist to find the minimum height the car must start at and still successfully negotiate the loop. What should be the physicist's answer? Write your answer as a function of d.

A 5.0 kg bucket of water resting on a horizontal surface is tied to a vertical rope. The rope of negligible mass passes through a frictionless pulley. The other end of the rope is attached to a 6.0 kg weight. The weight is dropped from a height of 2.5 m. Find the speed of the weight just before it hits the floor.

A simple Atwood machine is formed by two masses, m₁ and m₂, hanging over a massless pulley. A light cord connects the two masses. For a laboratory experiment, a student puts m₁ on the floor and holds with her hand m₂ at the height of H. The mass of m₂ is double that of m₁. The student releases the mass m₂ and measures with a motion detector the impact speed to be 2.6 m/s for m₂. From what height did the student release m₂?

A science teacher pushes a 0.75 kg cube against an ideal spring with a spring constant of 300 N/m. The spring is compressed to a distance of 5.0 cm. The teacher releases the cube. After moving over the smooth horizontal part of the setup shown in the figure below, the cube slides down over a rough ramp, making an angle of 15° with the horizontal. The coefficient of kinetic friction between the cube and ramp is 0.25. Determine the speed of the cube just after leaving contact with the spring.

Determine the maximum speed of oscillation of a nitrogen atom if the minimum potential energy of the nitrogen atom is 1.79 × 10^-19 J. Consider the mass of a nitrogen atom is 14u, where 1 u = 1 atomic mass unit = 1.66 × 10^-27 kg.

A 2.5 kg object is initially at rest at position s = 5.0 m on a potential energy diagram given by U(s) = -2s³ + 4s² J. Determine the object's velocity when it reaches position s = 8.0 m.

The figure below shows the process in which the work is done by the environment. Comment whether energy transfer occurs from system to environment or vice versa.

A snowboarder begins at rest and descends a friction-free snowy hill that tilts at a 9.00° angle with a vertical height of 116 m. What is their velocity upon reaching the bottom?

A toboggan is initially pushed up a friction-free slope set at a 17.0° angle, achieving a height of 1.23 m above its launch point. Determine its initial speed.

What minimum initial speed is required for a basketball player to dunk the ball by elevating their center of mass by 2.11 m without mechanical aids during their jump while still maintaining a speed of 0.51 m/s at the peak of their jump?

Find the thermal energy dissipated by the brakes of a 10,000 kg truck descending a 20.0° hill. The truck commences braking at 60.0 km/h, decelerating to 20.0 km/h over a 0.40 km stretch of road

A 63 kg gymnast jumps from the table with a vertical speed of 4.6 m/s directed upwards. Calculate the gymnast's speed as he safely reaches the bouncing mat, which is 2.1 m lower.

A 10.0-kg dog slides down a ramp 2.30 m high and arrives at the bottom from a stationary start, moving at 1.40 m/s. Determine the quantity of thermal energy generated by friction in this action.

A 4.0-g sparrow propels itself into the air, reaching 4.0 m/s speed in its ascent. Determine the sparrow's kinetic energy at this speed. Calculate the total energy expended in achieving this jump if the sparrow utilizes energy with 40% efficiency.

A 68-kg firefighter in training jumps upward from the top of a training structure with a speed of 4.0 m/s. A safety cushion, which can be likened to a spring with a spring constant of 5.5 x 10⁴ N/m, is set up 2.2 meters directly below the structure. Assess how much the cushion is compressed as the firefighter lands on it after the jump.

A wooden crate weighing 10 kg is positioned stationary at the top of the ramp as shown in the figure below. It is released from point P. Calculate the velocity of the crate at point Q.

A small cart of mass 2.0 kg is placed at the top of a smooth circular track DE of radius 3.0 m as illustrated below. Following DE, the cart encounters a horizontal section EF, where the coefficient of kinetic friction is 0.3. Subsequently, the track features a frictionless section FG where a cushion having spring like properties is situated. The cart, initially at rest at point D, is released to traverse the track. Upon reaching point F, the cart compresses the cushion. Determine the thermal energy produced as the cart moves from point E to point F.

A toy car of mass 1.5 kg is positioned at the top of a frictionless ramp (KL), depicted in the diagram below. The next part of the track is a horizontal span (LM) measuring 5.0 m with a coefficient of kinetic friction of 0.35. The subsequent section, MN, features a frictionless segment where a damping device is located. The car, initially stationary at point K, is released to traverse the track. Upon reaching point M, the car activates the damping device causing it to compress. Determine the velocity of the car at point M if the velocity of the car at point L is 6.2 m/s.

A toy truck, with a mass of 1.8 kg, is released from rest from the top of a sand hill as illustrated below. Upon reaching point Y, the toy truck compresses the spring-like device by 0.15 m. Calculate the stiffness constant k for the spring-like device if the velocity of the truck at point Y is 3.5 m/s and the portion YZ is frictionless.

An adventurer with a mass of 70 kg climbs from a valley at an elevation of 1.0 × 10³ m to reach the top of a cliff at an altitude of 2.6 × 10³ m. Calculate her change in gravitational potential energy in doing so.