Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Siyavula textbooks: Grade 10 Physical Science » Mechanical energy

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Bookshare

    This collection is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech Initiative

    Comments:

    "Accessible versions of this collection are available at Bookshare. DAISY and BRF provided."

    Click the "Bookshare" link to see all content affiliated with them.

  • FETPhysics display tagshide tags

    This module is included inLens: Siyavula: Physics (Gr. 10-12)
    By: Siyavula

    Review Status: In Review

    Click the "FETPhysics" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Siyavula: Physical Science display tagshide tags

    This collection is included inLens: Siyavula textbooks: Physical Science
    By: Free High School Science Texts Project

    Click the "Siyavula: Physical Science" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
Download
x

Download collection as:

  • PDF
  • EPUB (what's this?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(what's this?)" link.

  • More downloads ...

Download module as:

Reuse / Edit
x

Collection:

Module:

Add to a lens
x

Add collection to:

Add module to:

Add to Favorites
x

Add collection to:

Add module to:

 

Mechanical Energy

Tip:

Mechanical energy is the sum of the gravitational potential energy and the kinetic energy.

Mechanical energy, UU, is simply the sum of gravitational potential energy (PEPE) and the kinetic energy (KEKE). Mechanical energy is defined as:

U = P E + K E U = P E + K E
(1)
U = P E + K E U = m g h + 1 2 m v 2 U = P E + K E U = m g h + 1 2 m v 2
(2)

Conservation of Mechanical Energy

The Law of Conservation of Energy states:

Energy cannot be created or destroyed, but is merely changed from one form into another.

Definition 1: Conservation of Energy

The Law of Conservation of Energy: Energy cannot be created or destroyed, but is merely changed from one form into another.

So far we have looked at two types of energy: gravitational potential energy and kinetic energy. The sum of the gravitational potential energy and kinetic energy is called the mechanical energy. In a closed system, one where there are no external forces acting, the mechanical energy will remain constant. In other words, it will not change (become more or less). This is called the Law of Conservation of Mechanical Energy and it states:

The total amount of mechanical energy in a closed system remains constant.

Definition 2: Conservation of Mechanical Energy

Law of Conservation of Mechanical Energy: The total amount of mechanical energy in a closed system remains constant.

This means that potential energy can become kinetic energy, or vise versa, but energy cannot 'dissappear'. The mechanical energy of an object moving in the Earth's gravitational field (or accelerating as a result of gravity) is constant or conserved, unless external forces, like air resistance, acts on the object.

We can now use the conservation of mechanical energy to calculate the velocity of a body in freefall and show that the velocity is independent of mass.

Show by using the law of conservation of energy that the velocity of a body in free fall is independent of its mass.

Tip:

In problems involving the use of conservation of energy, the path taken by the object can be ignored. The only important quantities are the object's velocity (which gives its kinetic energy) and height above the reference point (which gives its gravitational potential energy).

Tip:

In the absence of friction, mechanical energy is conserved and
U before = U after U before = U after
(3)

In the presence of friction, mechanical energy is not conserved. The mechanical energy lost is equal to the work done against friction.

Δ U = U before - U after = work done against friction Δ U = U before - U after = work done against friction
(4)

In general, mechanical energy is conserved in the absence of external forces. Examples of external forces are: applied forces, frictional forces and air resistance.

In the presence of internal forces like the force due to gravity or the force in a spring, mechanical energy is conserved.

The following simulation covers the law of conservation of energy.
run demo

Figure 1
Figure 1 (energy-skate-park-screenshot.png)

Using the Law of Conservation of Energy

Mechanical energy is conserved (in the absence of friction). Therefore we can say that the sum of the PEPE and the KEKE anywhere during the motion must be equal to the sum of the PEPE and the KEKE anywhere else in the motion.

We can now apply this to the example of the suitcase on the cupboard. Consider the mechanical energy of the suitcase at the top and at the bottom. We can say:

Figure 2
Figure 2 (PG10C3_009.png)

U t o p = U b o t t o m P E t o p + K E t o p = P E b o t t o m + K E b o t t o m m g h + 1 2 m v 2 = m g h + 1 2 m v 2 ( 1 ) ( 9 , 8 ) ( 2 ) + 0 = 0 + 1 2 ( 1 ) ( v 2 ) 19 , 6 J = 1 2 v 2 39 , 2 = v 2 v = 6 , 26 m · s - 1 U t o p = U b o t t o m P E t o p + K E t o p = P E b o t t o m + K E b o t t o m m g h + 1 2 m v 2 = m g h + 1 2 m v 2 ( 1 ) ( 9 , 8 ) ( 2 ) + 0 = 0 + 1 2 ( 1 ) ( v 2 ) 19 , 6 J = 1 2 v 2 39 , 2 = v 2 v = 6 , 26 m · s - 1
(5)

The suitcase will strike the ground with a velocity of 6,26 m·s-16,26m·s-1.

From this we see that when an object is lifted, like the suitcase in our example, it gains potential energy. As it falls back to the ground, it will lose this potential energy, but gain kinetic energy. We know that energy cannot be created or destroyed, but only changed from one form into another. In our example, the potential energy that the suitcase loses is changed to kinetic energy.

The suitcase will have maximum potential energy at the top of the cupboard and maximum kinetic energy at the bottom of the cupboard. Halfway down it will have half kinetic energy and half potential energy. As it moves down, the potential energy will be converted (changed) into kinetic energy until all the potential energy is gone and only kinetic energy is left. The 19,6J19,6J of potential energy at the top will become 19,6J19,6J of kinetic energy at the bottom.

Exercise 1: Using the Law of Conservation of Mechanical Energy

During a flood a tree truck of mass 100kg100kg falls down a waterfall. The waterfall is 5m5m high. If air resistance is ignored, calculate

  1. the potential energy of the tree trunk at the top of the waterfall.
  2. the kinetic energy of the tree trunk at the bottom of the waterfall.
  3. the magnitude of the velocity of the tree trunk at the bottom of the waterfall.

Figure 3
Figure 3 (PG10C3_005.png)

Exercise 2: Pendulum

A 2kg2kg metal ball is suspended from a rope. If it is released from point AA and swings down to the point BB (the bottom of its arc):

  1. Show that the velocity of the ball is independent of it mass.
  2. Calculate the velocity of the ball at point BB.

Figure 4
Figure 4 (PG10C3_006.png)

Potential Energy

  1. A tennis ball, of mass 120g120g, is dropped from a height of 5m5m. Ignore air friction.
    1. What is the potential energy of the ball when it has fallen 3m3m?
    2. What is the velocity of the ball when it hits the ground?
    Click here for the solution
  2. A bullet, mass 50g50g, is shot vertically up in the air with a muzzle velocity of 200m·s-1200m·s-1. Use the Principle of Conservation of Mechanical Energy to determine the height that the bullet will reach. Ignore air friction.
    Click here for the solution
  3. A skier, mass 50kg50kg, is at the top of a 6,4m6,4m ski slope.
    1. Determine the maximum velocity that she can reach when she skies to the bottom of the slope.
    2. Do you think that she will reach this velocity? Why/Why not?
    Click here for the solution
  4. A pendulum bob of mass 1,5kg1,5kg, swings from a height A to the bottom of its arc at B. The velocity of the bob at B is 4m·s-14m·s-1. Calculate the height A from which the bob was released. Ignore the effects of air friction.
    Click here for the solution
  5. Prove that the velocity of an object, in free fall, in a closed system, is independent of its mass.
    Click here for the solution

Energy graphs

Let us consider our example of the suitcase on the cupboard, once more.

Figure 5
Figure 5 (PG10C3_010.png)

Let's look at each of these quantities and draw a graph for each. We will look at how each quantity changes as the suitcase falls from the top to the bottom of the cupboard.

  • Potential energy: The potential energy starts off at a maximum and decreases until it reaches zero at the bottom of the cupboard. It had fallen a distance of 2 metres.
    Figure 6
    Figure 6 (PG10C3_011.png)
  • Kinetic energy: The kinetic energy is zero at the start of the fall. When the suitcase reaches the ground, the kinetic energy is a maximum. We also use distance on the xx-axis.
    Figure 7
    Figure 7 (PG10C3_012.png)
  • Mechanical energy: The mechanical energy is constant throughout the motion and is always a maximum. At any point in time, when we add the potential energy and the kinetic energy, we will get the same number.
    Figure 8
    Figure 8 (PG10C3_013.png)

The following presentation provides a summary of some of the concepts covered in this chapter.

Figure 9

Summary

  • Mass is the amount of matter an object is made up of.
  • Weight is the force with which the Earth attracts a body towards its centre.
  • Newtons Law of Gravitation.
  • A body is in free fall if it is moving in the Earth's gravitational field and no other forces act on it.
  • The equations of motion can be used for free fall problems. The acceleration (a) is equal to the acceleration due to gravity (g).
  • The potential energy of an object is the energy the object has due to his position above a reference point.
  • The kinetic energy of an object is the energy the object has due to its motion.
  • Mechanical energy of an object is the sum of the potential energy and kinetic energy of the object.
  • The unit for energy is the joule (J).
  • The Law of Conservation of Energy states that energy cannot be created or destroyed, but can only be changed from one form into another.
  • The Law of Conservation of Mechanical Energy states that the total mechanical energy of an isolated system remains constant.
  • The table below summarises the most important equations:
Table 1
Weight F g = m · g F g = m · g
Equation of motion v f = v i + g t v f = v i + g t
Equation of motion Δ x = ( v i + v f ) 2 t Δ x = ( v i + v f ) 2 t
Equation of motion Δ x = v i t + 1 2 g t 2 Δ x = v i t + 1 2 g t 2
Equation of motion v f 2 = v i 2 + 2 g Δ x v f 2 = v i 2 + 2 g Δ x
Potential Energy P E = m g h P E = m g h
Kinetic Energy K E = 1 2 m v 2 K E = 1 2 m v 2
Mechanical Energy U = K E + P E U = K E + P E

End of Chapter Exercises: Gravity and Mechanical Energy

  1. Give one word/term for the following descriptions.
    1. The force with which the Earth attracts a body.
    2. The unit for energy.
    3. The movement of a body in the Earth's gravitational field when no other forces act on it.
    4. The sum of the potential and kinetic energy of a body.
    5. The amount of matter an object is made up of.
    Click here for the solution
  2. Consider the situation where an apple falls from a tree. Indicate whether the following statements regarding this situation are TRUE or FALSE. Write only 'true' or 'false'. If the statement is false, write down the correct statement.
    1. The potential energy of the apple is a maximum when the apple lands on the ground.
    2. The kinetic energy remains constant throughout the motion.
    3. To calculate the potential energy of the apple we need the mass of the apple and the height of the tree.
    4. The mechanical energy is a maximum only at the beginning of the motion.
    5. The apple falls at an acceleration of 9,8 m·s-29,8m·s-2.
    Click here for the solution
  3. [IEB 2005/11 HG] Consider a ball dropped from a height of 1m1m on Earth and an identical ball dropped from 1m1m on the Moon. Assume both balls fall freely. The acceleration due to gravity on the Moon is one sixth that on Earth. In what way do the following compare when the ball is dropped on Earth and on the Moon.
    Table 2
     MassWeightIncrease in kinetic energy
    (a)the samethe samethe same
    (b)the samegreater on Earthgreater on Earth
    (c)the samegreater on Earththe same
    (d)greater on Earthgreater on Earthgreater on Earth
    Click here for the solution
  4. A man fires a rock out of a slingshot directly upward. The rock has an initial velocity of 15 m·s-115m·s-1.
    1. How long will it take for the rock to reach its highest point?
    2. What is the maximum height that the rock will reach?
    3. Draw graphs to show how the potential energy, kinetic energy and mechanical energy of the rock changes as it moves to its highest point.
    Click here for the solution
  5. A metal ball of mass 200g200g is tied to a light string to make a pendulum. The ball is pulled to the side to a height (A), 10cm10cm above the lowest point of the swing (B). Air friction and the mass of the string can be ignored. The ball is let go to swing freely.
    1. Calculate the potential energy of the ball at point A.
    2. Calculate the kinetic energy of the ball at point B.
    3. What is the maximum velocity that the ball will reach during its motion?
    Click here for the solution
  6. A truck of mass 1,2tons1,2tons is parked at the top of a hill, 150m150m high. The truck driver lets the truck run freely down the hill to the bottom.
    1. What is the maximum velocity that the truck can achieve at the bottom of the hill?
    2. Will the truck achieve this velocity? Why/why not?
    Click here for the solution
  7. A stone is dropped from a window, 3m3m above the ground. The mass of the stone is 25g25g.
    1. Use the Equations of Motion to calculate the velocity of the stone as it reaches the ground.
    2. Use the Principle of Conservation of Energy to prove that your answer in (a) is correct.
    Click here for the solution

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Reuse / Edit:

Reuse or edit collection (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.

| Reuse or edit module (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.