A
metal ball is suspended from a rope. If it is released from point
and swings down to the point
(the bottom of its arc):
Show that the velocity of the ball is independent of it mass.
Calculate the velocity of the ball at point
.
The mass of the metal ball is
The change in height going from point
to point
is
The ball is released from point
so the velocity at point,
.
All quantities are in SI units.
Prove that the velocity is independent of mass.
Find the velocity of the metal ball at point
.
As there is no friction, mechanical energy is conserved. Therefore:
As the mass of the ball
appears on both sides of the equation, it can be eliminated so that the equation becomes:
This proves that the velocity of the ball is independent of its mass. It does not matter what its mass is, it will always have the same velocity when it falls through this height.
We can use the equation above, or do the calculation from 'first principles':
Potential energy
A tennis ball, of mass
, is dropped from a height of
. Ignore air friction.
What is the potential energy of the ball when it has fallen
?
What is the velocity of the ball when it hits the ground?
A bullet, mass
, is shot vertically up in the air with a muzzle velocity of
. Use the Principle of Conservation of Mechanical Energy to determine the height that the bullet will reach. Ignore air friction.
A skier, mass
, is at the top of a
ski slope.
Determine the maximum velocity that she can reach when she skies to the bottom of the slope.
Do you think that she will reach this velocity? Why/Why not?
A pendulum bob of mass
, swings from a height A to the bottom of its arc at B. The velocity of the bob at B is
. Calculate the height A from which the bob was released. Ignore the effects of air friction.
Prove that the velocity of an object, in free fall, in a closed system, is independent of its mass.
Energy graphs - (not in caps, included for completeness)
Let us consider our example of the suitcase on the cupboard, once more.
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.
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
-axis.
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.
Summary
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:
Potential Energy
Kinetic Energy
Mechanical Energy
End of chapter exercises: gravity and mechanical energy
Give one word/term for the following descriptions.
The force with which the Earth attracts a body.
The unit for energy.
The movement of a body in the Earth's gravitational field when no other forces act on it.
The sum of the potential and kinetic energy of a body.
The amount of matter an object is made up of.
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.
The potential energy of the apple is a maximum when the apple lands on the ground.
The kinetic energy remains constant throughout the motion.
To calculate the potential energy of the apple we need the mass of the apple and the height of the tree.
The mechanical energy is a maximum only at the beginning of the motion.
The apple falls at an acceleration of
.
A man fires a rock out of a slingshot directly upward. The rock has an initial velocity of
.
What is the maximum height that the rock will reach?
Draw graphs to show how the potential energy, kinetic energy and mechanical energy of the rock changes as it moves to its highest point.
A metal ball of mass
is tied to a light string to make a pendulum. The ball is pulled to the side to a height (A),
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.
Calculate the potential energy of the ball at point A.
Calculate the kinetic energy of the ball at point B.
What is the maximum velocity that the ball will reach during its motion?
A truck of mass
is parked at the top of a hill,
high. The truck driver lets the truck run freely down the hill to the bottom.
What is the maximum velocity that the truck can achieve at the bottom of the hill?
Will the truck achieve this velocity? Why/why not?
A stone is dropped from a window,
above the ground. The mass of the stone is
.
Use the Principle of Conservation of Energy to determine the speed with which the stone strikes the ground.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product