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Wize University Physics Textbook (Master) > Work and Energy

Conservation of Energy

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Energy Conservation



The total mechanical energy of a system is the sum of kinetic and potential energies at a given time.

Remember that Wnet=ΔKW_{net}=\Delta Kusing Work-Energy Theorem. We also remember that for conservative force, Wc=−ΔUW_c=-\Delta U. Hence we have:



ΔK=Wnet=Wc+Wnc=−ΔU+Wnc\Delta K=W_{net}=W_c+W_{nc}=-\Delta U+W_{nc}



Solving for WncW_{nc} we will find:


Wnc= ΔK + ΔU=ΔE\boxed{W_{nc}=\ \Delta K\ +\ \Delta U=\Delta E}

where WncW_{nc} is the total work done by non-conservative forces and ΔE\Delta E is the total change in the mechanical energy

Watch Out!
WncW_{nc} could be positive, negative or zero:
  • If it is positive, the mechanical energy is increased (such as pulling force)
  • If it is negative, the mechanical energy is decreased (such as friction)
  • if it is zero. energy is conserved

The above equation sometimes is written is the following form:



Wadded+Ki+Ui=Kf+Uf+WlostW_{added}+K_{i}+U_{i}=K_{f}+U_{f}+W_{lost}



where iiand ffindexes refer to initial and final situations respectively. Furthermore, here WaddedW_{added} and WlostW_{lost} are positive and negative non-conservative works respectively.


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Energy Conservation

If there is no non-conservative work, and there is no dissipated energy, there is no change in energy:

ΔE = 0  →  ΔK +ΔU=0\Delta E\ =\ 0\ \ \rightarrow \ \ \Delta K\ + \Delta U=0

- This is the conservation of energy: energy is not destroyed or created, but transformed into different forms.


Exam Tip
We can use conservation of energy to solve dynamics problem when we are interested in knowing initial and final conditions, without being too concerned about what happens exactly in between.









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Example: Work Done by Friction


A car parked on a 32- degree steep hill loses its traction and starts to slide down. It travels 160 meters down the hill until it reaches speed of 72 km/h. What is the kinetic coefficient of friction between the car and the surface?


Solution:

The car at the top is at rest, while at the bottom, it reaches the velocity of 72 km/h (20 m/s). The only non conservative force that does work on the car is the friction of the surface which does a negative work. So, the mechanical energy is not conserved.

Thus:
Wnc=Ef−EiW_{nc}=E_f-E_i

Wfric=(K+Ug)f−(K+Ug)iW_{fric}=\left(K+U_g\right)_f-\left(K+U_g\right)_i

fk×d×(−1)=(K+0)f−(0+Ug)if_k\times d\times\left(-1\right)=\left(K+0\right)_f-\left(0+U_g\right)_i


Here we set the zero of gravitational potential energy at the end of the path. Note the negative sign of friction work due to cos⁡(180°)\cos(180\degree)
Furthermore, kinetic friction can be written as: fk=μkN=μkmgcos⁡(32)f_k=\mu_kN=\mu_k mg\cos(32)

μk(mg×cos⁡(32))×d×(−1)=12mvf2−mghi\mu_k\left(mg\times\cos\left(32\right)\right)\times d\times\left(-1\right)=\frac{1}{2}mv_f^2-mgh_i
This is a very important step. What we know is that the car has travelled d meters down the hill. But to find hih_iwe need to find the vertical change in the position using a little bit of geometry:
μk(mg×cos⁡(32))×d×(−1)=12mvf2−mg(d sin⁡(32))\mu_k\left(mg\times\cos\left(32\right)\right)\times d\times\left(-1\right)=\frac{1}{2}mv_f^2-mg\left(d\ \sin\left(32\right)\right)
Mass is cancelled from both sides of equations. Thus:
μk(g×cos⁡(32))×d×(−1)=12(20)2−g(d sin⁡(32))\mu_k\left(g\times\cos\left(32\right)\right)\times d\times\left(-1\right)=\frac{1}{2}\left(20\right)_{ }^2-g\left(d\ \sin\left(32\right)\right)

μk(9.81×cos⁡(32))×(160)×(−1)=12(400)−(9.81)(160)(sin⁡(32))\mu_k\left(9.81\times\cos\left(32\right)\right)\times\left(160\right)\times\left(-1\right)=\frac{1}{2}\left(400\right)-\left(9.81\right)\left(160\right)\left(\sin\left(32\right)\right)

μk=0.47\mu_k=0.47




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A 2.00-kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220 m. When the block is released , it moves along a friction-less , horizontal surface and then up a friction-less incline with slope 37.0∘37.0^{\circ}

(a) What is the speed of the block as it slides along the horizontal surface after having lost the contact with the spring?
(b) How far does the block travel up the incline before starting to slide back down?




Extra Practice
conservation of energy
Practice: Conservation of Energy
A 2 kg2\ kgboy is dropped by firefighters from a height of 10 m10\ m to be caught by a team of rescuers on the ground. Determine the velocity of the boy before he reaches the ground level.
work and energy
A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to
(a) the ground
(b) and the top of the person’s head?
Problems on Work and Energy
A 2.00-kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220 m. When the block is released , it moves along a friction-less , horizontal surface and then up a friction-less incline with slope 37.0∘37.0^{\circ}(Fig).
(a) What is the speed of the block as it slides along the horizontal surface after having left the spring?
(b) How far does the block travel up the incline before starting to slide back down?
Problems on Work and Energy
A 2.00-kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220 m. When the block is released , it moves along a friction-less , horizontal surface and then up a friction-less incline with slope 37.0∘37.0^{\circ}(Fig).
(a) What is the speed of the block as it slides along the horizontal surface after having left the spring?
(b) How far does the block travel up the incline before starting to slide back down?
work and energy
A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to
(a) the ground
(b) and the top of the person’s head?
Example: Releasing two repulsive charges (harder)
EX: Two point charges are 20 cm apart and each carry 3 µc of positive charge. At a moment, they are released to move along connecting line, repelling each other. What is the velocity of the charges when they are 60 cm apart? Consider the mass of the charges to be 0.3 grams. Give your answer in m/s.
Graphical problem
Practice: Graphical Problem of Conservation of Energy
A 1.2-g particle is moving along x-axis under the influence of the following potential energy. The particle turns around at x=6mx=6m. What is the maximum speed of this particle?
Problems on Momentum Conservation and Collisions
Practice: Conservation of Momentum with Spring
A 15.0-kg block is attached to a very light horizontal spring with spring constant 500 N/m and is resting on a friction-less horizontal table. It is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right. The stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision. Enter the answer with 3 significant digits.
Problems on Work and Energy
Practice: Block with Spring and Ramp
A 2.00-kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220 m. When the block is released , it moves along a friction-less , horizontal surface and then up a friction-less incline with slope 37.0∘37.0^{\circ}(Fig).
(a) What is the speed of the block as it slides along the horizontal surface after having left the spring?
conservation of energy
Practice: Conservation of Energy
A 2 kg2\ kgboy is dropped by firefighters from a height of 10 m10\ m to be caught by a team of rescuers on the ground. Determine the velocity of the boy before he reaches the ground level.
conservation of energy
Practice: Conservation of Energy
A 2 kg2\ kgboy is dropped by firefighters from a height of 10 m10\ m to be caught by a team of rescuers on the ground. Determine the velocity of the boy before he reaches the ground level.
Releasing two repulsive charges
Two point charges are held 20 cm apart and each carries 3.0 µc of positive charge. At the same time, they are released, and they begin to move straight apart from each other. What is the velocity of the charges when they are 60 cm apart? Consider the mass of the charges to be 0.3 grams. Give your answer in m/s.
conservation of energy
Practice: Conservation of Energy
A 2 kg2\ kgboy is dropped by firefighters from a height of 10 m10\ m to be caught by a team of rescuers on the ground. Determine the velocity of the boy before he reaches the ground level.
Practice: Spring constant
A 300-gram ball starts to roll down 4 meters on a 30 degree incline. (That is, the distance it travels along the slope is 4 meters.) The ball then rolls 2 meters on flat surface and hits a plate, connected to a spring. The spring is compressed 25 cm. If the kinetic coefficient of friction of all surfaces is 0.1, what is the spring constant?
Conservation of energy
A 1.2-g particle is moving along x-axis under the influence of the following potential energy. The particle turns around at x=6mx=6m. What is the maximum speed of this particle?
A 2 kg2\ kgball is dropped from a height of 10 m10\ monto the ground.
a) If the balls experiences no losses find the velocity of the ball right before it hits the ground. (in m/s)
b) If the ball loses 25% of its energy when it bounces, what height will it reach on its first bounce? (in m)
Two balls of equal mass are thrown from the top of a building with height10.0m10.0 m. One is thrown straight up at 10 m/s10\ m/s(call this ball A) and the other is thrown horizontally at 10 m/s10\ m/s (call this ball B).
a) Which balls has a greater speed when it reaches the ground? Why? Make sure you include conservation of energy in your answer.
b) Your friend tries to tell you that the ball thrown horizontally would have greater speed, because it has both an x-component and a y-component, and so the speed (magnitude of the velocity) should be higher. Why is he wrong? (Calculation not required, but you can do one if you want.)
A marble starts from rest and rolls without slipping on the loop-the-loop track in the figure. Find the minimum starting height h from which the ball will remain on the track throughout the loop. Assume the marble's radius is small compared with R. The moment of inertia for a solid sphere is 25Mr2\frac25Mr^2.
Find your answer by using R = 70 cm (radius of the loop) and r = 1.0 cm (radius of the marble).
The system below initially starts from rest, with a spring constant of 28 N/m.
At t = 0, the 3 kg block is removed. If the 2 kg block is not attached to the spring:
A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless, smooth incline, as shown in the figure, and eventually stops after reaching a certain height.
If we repeat this experiment but instead compress the spring a distance of 2x, what will happen? Explain your answer.
A spring with a spring constant k=340 N/m is used to launch a 1.5kg block along the horizontal surface whose coefficient of kinetic friction is 0.27. If the spring is compressed 18cm, how far does the block slide?
A frictionless roller-coaster track is guiding a car from point A to point D as shown below. The car at A is at rest and slowly starts its journey. Which set of energy diagrams does correctly represent the energy at indicated points?
A mass is on a spring. The spring is stretched and then released so that the mass bounces back and forth. There are no other forces in the system. At what point is the block moving the fastest?
A 20.0kg cart is moving at 10.0m/s before it goes down a hill. It ends up 2.0m below where it started, and travelled a distance of 15.0m. What is the speed of the car? There is friction, and it applies an average force of 2.00N over the course of it's motion.
Give your answer in m/s.
Problems on Work and Energy
A 2.00-kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220 m. When the block is released , it moves along a friction-less , horizontal surface and then up a friction-less incline with slope 37.0∘37.0^{\circ}(Fig).
(a) What is the speed of the block as it slides along the horizontal surface after having left the spring?
(b) How far does the block travel up the incline before starting to slide back down?
Graphical problem
A 1.2-g particle is moving along x-axis under the influence of the following potential energy. The particle turns around at x=6mx=6m. What is the maximum speed of this particle?
Problems on Momentum Conservation and Collisions
A 15.0-kg block is attached to a very light horizontal spring with spring constant 500 N/m and is resting on a friction-less horizontal table. It is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right. The stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision. Enter the answer with 3 significant digits.
Two cylinders of the same size but different masses roll down an incline, starting from rest. Cylinder A has a greater mass. Which reaches the bottom first?
conservation of energy
A 2 kg2\ kg ball is dropped from a height of 10 m10\ m onto the ground. If the balls experiences no losses find the velocity of the ball right before it hits the ground. If the ball loses 25%25\% of its energy when it bounces, what height will it reach on its first bounce?
work and energy
A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to
(a) the ground
(b) and the top of the person’s head?
A box of mass m is pressed against (but is not attached to ) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment but instead compress the spring a distance of 2x.
A horizontal rod of length L and mass M has a mass m at one end. It is supported by pivot P on the wall at the left end and a cable at angle θ\theta of at the other end as shown.
a) Find the tension in the cable
b) If the cable snaps, with what angular velocity will the rod swing down and slam into the wall?
A pendulum with length of 0.80m is raised to an angle of 25° below the horizontal and released. What is the velocity of the pendulum as it swings past its lowest point?
A person with mass of 75kg is doing bungee jumping such that he jumps down from rest form the top of a bridge. If the natural length of the bungee cord is 20m and effective spring constant for this bungee cord is 5N/m, what is the speed of the person 20m below the bridge? What is the lowest position of the person respect to the bridge?
A marble starts from rest and rolls without slipping on the loop-the-loop track in the figure. Find the minimum starting height h from which the ball will remain on the track through the loop. Assume the marble's radius is small compared with R.
A ball of mass M is released from rest at a vertical height h above the ground. The ball rolls down a frictionless inclined plane and makes it to a loop the loop of radius R.
What is the velocity of the ball as it reaches to the bottom of the incline?
What is the normal force at the bottom?
A 200g mass block is released from rest at a height of 25 cm on 30o incline. The incline is frictionless. It slides down the incline and then a frictionless surface until it collides elastically with an 800g block at rest 1.4 m from the bottom of the incline as shown in the figure. How much later do the two blocks collide again?
A ball of mass 0.5 kg is attached to a string of length 26 cm. The ball is raised so that the string is taut and horizontal. Now the ball is released such that it swings and undergoes an elastic head-on collision with a stationary ball of mass 0.20kg that is free to roll horizontally. a) Find the speed of the swinging ball just before the collision. b) What is the speed of the 0.20 kg ball just after the collision?
A car accidentally enters a playing ground at a velocity of 14m/s. A child tosses a ball at the car at 18m/s. What is the ball's speed after it rebounds elastically from the car?
A ballistic pendulum is used to measure the speed of a bullet. It consists of a wooden block of mass M = 10kg suspended from a vertical string. A bullet of mass m = 100g strikes and embeds itself in the block and the block swings upward through a vertical distance of 0.10m. What is the bullet's speed?
A 12 g bullet moving horizontally at 30 m/s is fired into a vertically suspended pendulum of mass 2.5 kg. The bullet becomes embedded in the pendulum.
As shown below, a ball of mass m with an initial velocity 10 m/s collides with an initially stationary ballistic pendulum of mass M=3m, hanging from a string. The two sticks together and system deflects from vertical position to angleΘ=60∘\Theta=60^{^{\circ}} What is the length of string?
Mechanical energy
When a rollercoaster cart moves towards the ground, what happens to the PE that it has?
A block of mass m is released from rest at the top of a semi- circular frictionless track of radius R.
Which of the following expressions most accurately represents the speed of the block at the bottom of the track?
g is the acceleration due to gravity.
You drop a 5 kg ball from a height of 2 m. Just before it reaches the ground, how much kinetic energy ( in N) does it have?
Mechanical energy
An object is lifted to some height and then dropped. During the drop, which of the following is increased? Neglect air resistance.
Releasing two repulsive charges
Two point charges are 20 cm apart and each carry 3 µc of positive charge. At a moment, they are released to move along connecting line, repelling each other. What is the velocity of the charges when they are 60 cm apart? Consider the mass of the charges to be 0.3 grams. Give your answer in m/s.
A 65 kg man is riding a Ferris wheel with radius of 15 m, that moves at 1.5 rpm. Determine the normal force that the man experiences at the top of the ferris wheel and at the bottom.
Conservation of Energy & Momentum
A 12-g rifle bullet is fired with a speed of 380 m/s into a ballistic pendulum with mass 6.00 kg, suspended from a cord 70.0 cm long. Compute (a) the vertical height through which the pendulum rises , (b) the initial kinetic energy of the bullet, and (c) the kinetic energy of the bullet and pendulum immediately after the bullet becomes embedded in the pendulum.
A 15.0-kg block is attached to a very light horizontal spring of force constant 500 N/m and is resting on a friction-less horizontal table (Fig.). Suddenly it is struck by a 3.00-kg stone traveling horizontally at 8.00 m/s to the right, where upon the stone rebound at 2.00 m/s horizontally to the left.Find the maximum distance that the block will compress the spring after the collision.
A 5 kg chunk of ice is sliding at 12.0 m/s on the floor of an ice-covered valley when it collides with and stick to another 5.00-kg chunk of ice that is initially at rest.(Fig.) Since the valley is icy, there is no friction.After the collision, how high above the valley floor will the combined chunks go?
A 0.15 kg frame, when suspended from a coil spring, stretches the spring 0.070 m.A 0.200-kg lump of putty is dropped from rest onto the frame from a height of 30.0 cm(Fig). Find the maximum distance the frame moves downward from its initial position.
Two identical masses are released from rest in a smooth hemispherical bowl of radius R from the positions shown in Fig. You can ignore friction between the masses and the surface of bowl. If they stick together when they collide, how high above the bottom of the bowl will the masses go after colliding in terms of R?
A ball is thrown from the top of a 10m tall building with a speed of 5m/s. What is its speed as it hits the ground?
A frictionless roller-coaster track is guiding a car from point A to point D as shown below. The car at
A is at rest and slowly starts its journey. Which set of energy diagrams does correctly represent the
energy at indicated points?
A 1.2-g particle is moving along x-axis under the influence of the following potential energy. The particle turns around at x=6mx=6m. What is the maximum speed of this particle?
A compressed spring is used to propel a 25-gram ball-bearing along a horizontal track which contains a circular vertical loop of radius 0.30 m in a vertical plane. The spring obeys Hooke's law and is compressed 20 cm a force of 5 N. What is the speed of the ball-bearing at the top of the loop?