Conservation of Mechanical Energy

A stone of $2\ \text{kg}$ falls from a cliff of $20\ \text{m}$ (take $g = 10\ \text{m/s}^2$ and ignore air resistance). The situation is shown in the figure.

1. a) What potential energy does it have when it starts to fall?
2. b) If mechanical energy is conserved, what kinetic energy will it have when it reaches the bottom?
3. c) At what speed will it reach the bottom?

You throw a ball of $0.5\ \text{kg}$ vertically upwards with a speed of $10\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$ and ignore friction), as shown in the figure.

1. a) What kinetic energy does it have when it leaves your hand?
2. b) Apply the conservation of mechanical energy between the release point and the highest point (where the speed is zero) and calculate the maximum height reached.

A skier goes down a slope, without pushing off, from a peak to the base (ignore friction), as shown in the figure.

1. a) At which point of the descent is their speed maximum? Where is their potential energy maximum?
2. b) Does their mechanical energy change during the descent?
3. c) If another slope with a completely different profile connected the same peak to the same base, would they reach the bottom with a higher, lower, or equal speed? Justify your answer using conservation of energy.

On all roller coasters, the first peak of the track is the highest of the entire circuit. Look at the diagram:

Explain, using the conservation of mechanical energy, why no subsequent peak can exceed the height of the first peak if the carriage starts from it almost from rest, and what role friction plays in that conclusion.

A roller coaster carriage passes the highest peak of the track, at $40\ \text{m}$, with a speed of $5\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$ and ignore friction). The situation is shown in the figure.

1. a) Calculate its mechanical energy per kilogram of mass.
2. b) At what speed will it pass through a point located at a height of $10\ \text{m}$?
3. c) Is it necessary to know the mass of the carriage to answer? Justify your answer using the conservation equation.

A laboratory trolley enters the base of a curved track with a speed of $4\ \text{m/s}$ and travels up it without pushing off (take $g = 10\ \text{m/s}^2$ and ignore friction).

1. a) What maximum height does it reach?
2. b) If it entered at twice the speed, $8\ \text{m/s}$, would it reach twice the height? Calculate it and comment on the result.
3. c) Write down the proportional relationship between the height reached and the entry speed.

A roller coaster carriage starts from rest at a peak $45\ \text{m}$ high, descends to a dip at $25\ \text{m}$ and then heads up a second peak of $40\ \text{m}$ (take $g = 10\ \text{m/s}^2$ and ignore friction). The profile of the track is shown in the figure.

1. a) What speed does it pass through the dip with?
2. b) What speed does it pass over the second peak with?
3. c) If there were a third peak at $50\ \text{m}$, would it manage to clear it? Justify your answer using energies and state the minimum height it should start from to achieve it (in the limit, arriving with zero speed).
4. d) Taking into account that in reality there is always friction, explain whether the starting height should be greater or smaller than the one calculated in the previous section.

A roller coaster carriage starts from rest at a peak of $45\ \text{m}$ and enters a loop of radius $12.5\ \text{m}$, whose highest point is therefore $25\ \text{m}$ above the ground (take $g = 10\ \text{m/s}^2$ and ignore friction). The diagram is shown in the figure.

1. a) At what speed does it pass through the lowest point of the circuit?
2. b) At what speed does it pass through the highest point of the loop?
3. c) For safety, the speed at the highest point of the loop must be at least $15\ \text{m/s}$ (given). Does the carriage exceed this? Calculate the minimum height from which the carriage could start in order to just satisfy this requirement.
4. d) Explain why in the real design the starting peak is well above this theoretical minimum height.

A pogo stick has a spring that compresses on landing and stretches on take-off.

1. a) Describe the sequence of energy transformations in a complete jump, from the moment the spring is fully compressed until the jumper reaches maximum height.
2. b) Is mechanical energy conserved while the spring pushes? What about during flight? Reason both answers.

A spring of $k = 200\ \text{N/m}$, compressed by $0.1\ \text{m}$, pushes a $0.5\ \text{kg}$ block initially at rest across a frictionless horizontal table. The diagram is shown in the figure.

1. a) What elastic potential energy does the compressed spring have?
2. b) Apply the conservation of energy between the compressed spring and the block once separated from the spring, and calculate the speed at which it is launched.
3. c) Explain what forces act on the block at each stage and why mechanical energy is conserved.

A mass of $1\ \text{kg}$ hangs from a vertical spring of $k = 100\ \text{N/m}$. It is released from rest at the point where the spring is unstretched and falls, stretching it (take $g = 10\ \text{m/s}^2$ and ignore friction). The diagram is shown in the figure.

1. a) Explain why, both when released and at the lowest point of the fall, its kinetic energy is zero, and set up the conservation of energy between these two points: $m \times g \times x = \frac{1}{2} \times k \times x^2$, where $x$ is the maximum extension.
2. b) Calculate the maximum extension $x$.
3. c) The static equilibrium position (where the hanging mass would remain at rest) satisfies $k \times x_e = m \times g$: calculate it and compare it with $x$. Explain the factor $2$ that appears: where is the 'extra energy' when the mass passes through the equilibrium position?

A $2\ \text{kg}$ package slides down a metal ramp from a height of $5\ \text{m}$ and reaches the bottom with a speed of $8\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$). The diagram is shown in the figure.

1. a) What potential energy did it have at the top?
2. b) What kinetic energy does it have when it reaches the bottom?
3. c) Has mechanical energy been conserved? Calculate the energy dissipated, which equals the magnitude of the work done by the friction force, $W_{roz} = \Delta E_m$.
4. d) If the ramp is $10\ \text{m}$ long, calculate the average friction force acting on the package.

A box of $4\ \text{kg}$ slides across a horizontal floor with an initial speed of $6\ \text{m/s}$ and comes to a stop due to friction. The coefficient of friction is $\mu = 0.3$ (take $g = 10\ \text{m/s}^2$).

1. a) What kinetic energy did it have when it started to slide?
2. b) Calculate the friction force, $F_{roz} = \mu \times m \times g$.
3. c) The work done by friction dissipates all the kinetic energy: calculate the distance it travels before coming to a stop.

A ball of $0.1\ \text{kg}$ is dropped from $2\ \text{m}$ and, after bouncing on the ground, only reaches a height of $1.5\ \text{m}$ (take $g = 10\ \text{m/s}^2$), as shown in the figure.

1. a) Calculate its potential energy before falling and at its highest point after the bounce.
2. b) How much energy has it lost in the bounce? What percentage of the initial energy does this represent?
3. c) What has this energy been transformed into?
4. d) If the ball continued bouncing in the same way, what would eventually happen to its mechanical energy?

The $0.2\ \text{kg}$ bob of a pendulum passes through the lowest point with a speed of $4\ \text{m/s}$ (take $g = 10\ \text{m/s}^2$).

1. a) What kinetic energy does it have at that point?
2. b) If there were no friction, what maximum height would it reach on the other side?
3. c) In reality it only reaches $0.6\ \text{m}$: calculate the energy dissipated in this half-oscillation, $W_{roz} = \Delta E_m$.
4. d) Explain, based on these numbers, why the pendulum will eventually stop and what its energy has been converted into.

A spring of $k = 500\ \text{N/m}$, compressed by $0.2\ \text{m}$, launches a $1\ \text{kg}$ block across a horizontal floor with a coefficient of friction of $\mu = 0.25$ (take $g = 10\ \text{m/s}^2$). The block travels a certain distance and comes to a stop.

1. a) Calculate the initial elastic potential energy of the spring.
2. b) Calculate the friction force acting on the block once it is moving.
3. c) Explain why all the elastic energy is eventually dissipated and calculate the distance the block travels before stopping.
4. d) If the spring were compressed by $0.4\ \text{m}$ (twice as much), what distance would it travel? Comment on the relationship you find.

A car of $1200\ \text{kg}$ brakes to a complete stop on dry asphalt, with a coefficient of friction of $\mu = 0.6$ (take $g = 10\ \text{m/s}^2$).

1. a) Calculate the friction force during braking.
2. b) If it was travelling at $30\ \text{m/s}$, calculate its kinetic energy and the braking distance (the work done by friction consumes all the kinetic energy).
3. c) Repeat the calculation for an initial speed of $15\ \text{m/s}$.
4. d) What is the relationship between the two braking distances? Explain the consequence of this result for urban speed limits.

A block of $2\ \text{kg}$ slides down from rest along an inclined plane of height $3\ \text{m}$. The angle of the plane satisfies $\sen \theta = 0.6$ and $\cos \theta = 0.8$, and the coefficient of friction is $\mu = 0.3$ (take $g = 10\ \text{m/s}^2$). The diagram is shown in the figure.

1. a) Calculate the length of the plane covered by the block and the normal force acting on it, $N = m \times g \times \cos \theta$.
2. b) Calculate the friction force and its work done, $W_{roz} = -F_{roz} \times L$.
3. c) Apply $W_{roz} = \Delta E_m$ to calculate the kinetic energy of the block when reaching the bottom and its speed.
4. d) What percentage of the initial potential energy has been dissipated by friction?

A person and their equipment, totalling $80\ \text{kg}$, ride a zip line with $100\ \text{m}$ of cable that covers a height difference of $45\ \text{m}$. The average friction (pulley, air resistance) is $200\ \text{N}$ (take $g = 10\ \text{m/s}^2$). The diagram is shown in the figure.

1. a) What potential energy do they have at the start of the ride?
2. b) Calculate the work done by friction during the entire descent, $W_{roz} = -F_{roz} \times d$.
3. c) Calculate the kinetic energy and the speed with which they reach the end of the cable.
4. d) At what speed would they arrive if there were no friction? Comment on the result and explain why zip lines need a braking system at the end.

A swing swings freely after an initial push, as shown in the figure.

1. a) At which points along the motion is its potential energy maximum? And where is its kinetic energy maximum?
2. b) Describe the energy transformation that takes place as it descends from one extreme to the lowest point.
3. c) If no one pushes it again, the amplitude gradually decreases: which forces are responsible, and what happens to the mechanical energy?

State in which of these situations mechanical energy is conserved and in which it is not, justifying your answer with a sentence in each case:

1. a) A satellite in a circular orbit around the Earth.
2. b) A car braking hard to a complete stop.
3. c) An apple falling from a branch (ignore friction).
4. d) A lift ascending at a constant speed.

Explain in your own words what a conservative force is, using weight and the elastic force of a spring as examples, and what happens to the mechanical energy of a system when only forces of this type act. Then give an example of a non-conservative force, state the sign of the work it does on a moving object, and explain its effect on the object's mechanical energy.

In the laboratory, the $0.1\ \text{kg}$ bob of a pendulum is released from rest at a position $45\ \text{cm}$ higher than the lowest point of its swing (take $g = 10\ \text{m/s}^2$ and ignore friction), as shown in the figure.

1. a) Calculate its initial potential energy relative to the lowest point.
2. b) Apply the conservation of energy and calculate the speed with which it passes through the lowest point.
3. c) What height will it reach on the other side of the swing? Justify your answer without performing further calculations.

A $2\ \text{kg}$ package drops from a stationary delivery drone at a height of $45\ \text{m}$ (take $g = 10\ \text{m/s}^2$ and ignore air resistance).

1. a) What is its mechanical energy when it starts to fall?
2. b) As it passes a height of $25\ \text{m}$, how much potential energy and how much kinetic energy does it have? At what speed does it pass through there?
3. c) At what speed does it reach the ground?
4. d) Do the calculated speeds depend on the mass of the package? Justify your answer using the conservation equation.