The principle of conservation: mechanical energy in everyday situations

The pendulum of an old clock swings between its two highest positions, as shown in the figure.

1. a) At which points along its path is its potential energy maximum? And at which points is its kinetic energy maximum?
2. b) After a few days, the pendulum stops if no one winds it up: explain what role friction plays and what the clock mechanism (the weights) is used for.

A coin falls down a vertical well and reaches the bottom at $14\ \text{m/s}$, as shown in the figure. Assume there is no friction and take $g = 9,8\ \text{m/s}^2$. State the conservation of mechanical energy and calculate the depth of the well.

A parcel of $2\ \text{kg}$ goes down a metal ramp from a height of $3\ \text{m}$ and reaches the bottom at $6\ \text{m/s}$, as shown in the figure (take $g = 10\ \text{m/s}^2$).

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 dissipated energy and explain where it has gone.

A sandbag of $2\ \text{kg}$ falls from a stationary hot-air balloon at a height of $45\ \text{m}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) What is its mechanical energy when it starts falling?
2. b) When it passes a height of $25\ \text{m}$, how much potential energy and how much kinetic energy does it have?
3. c) At what speed does it pass through there?
4. d) At what speed does it reach the ground?

You drop a ball from the top of a roof and, at the same time, allow an identical ball to roll down a frictionless inclined plane starting from the same height, as shown in the figure.

1. a) Which one reaches the ground with greater speed? Reason your answer using the conservation of energy.
2. b) Which one arrives first? Explain the difference between final speed and time of descent.

A block of $3\ \text{kg}$ slides down a ramp from a height of $2\ \text{m}$ and then drags along the horizontal ground for $4\ \text{m}$, until it comes to a complete stop, as shown in the figure (take $g = 10\ \text{m/s}^2$).

1. a) What potential energy did it have at the top?
2. b) Reason why all that energy ends up being dissipated by friction.
3. c) Assuming that almost all of it is dissipated on the horizontal section, calculate the average friction force slowing down the block.

A roller coaster carriage starts from rest at a peak $50\ \text{m}$ high, falls into a valley located at $5\ \text{m}$ and then approaches a second peak of $45\ \text{m}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) At what speed does it pass through the valley?
2. b) At what speed does it pass over the second peak?
3. c) If there were a third peak of $55\ \text{m}$, would it manage to clear it? Justify your answer using energy considerations and state the minimum height it should start from to achieve this.

You throw a ball of $0,2\ \text{kg}$ vertically upwards with a speed of $15\ \text{m/s}$, as shown in the figure (take $g = 10\ \text{m/s}^2$, frictionless).

1. a) What kinetic energy does it have when it leaves your hand?
2. b) What maximum height will it reach? State the conservation of energy.

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

1. a) A skydiver descends with an open parachute at a constant speed.
2. b) The Space Station orbits around the Earth.
3. c) A van brakes while going down a mountain pass.

A ball of $0,4\ \text{kg}$ is dropped from $1,5\ \text{m}$ and, after bouncing on the ground, only rises to $0,9\ \text{m}$, as shown in the figure (take $g = 10\ \text{m/s}^2$).

1. a) Calculate its potential energy before falling and after bouncing.
2. b) How much energy has it lost in the bounce? What percentage of the initial energy is this?
3. c) What has that energy been transformed into?

A car of $1000\ \text{kg}$ travelling at $20\ \text{m/s}$ brakes to a complete stop; its brakes can exert an average force of $8000\ \text{N}$.

1. a) Calculate its kinetic energy and the minimum braking distance (the work done by the braking force consumes all the kinetic energy).
2. b) Repeat the calculation for an initial speed of $40\ \text{m/s}$.
3. c) What is the relationship between the two distances? Explain what consequence this has for road safety.