Work and Energy

Work and Energy

Learning Objectives:

1. Explain Energy.
2. Differentiate between kinetic energy and potential energy.
3. Show how the formula for kinetic energy is derived from the work formula.
4. Demonstrate how the formula for potential energy is derived from the work formula.
5. Prove how these formulas are used for answering problems.
   

ENERGY

Energy is commonly define as the ability to do work.

Two types of energy are Potential energy and Kinetic energy.

Example

• The energy in gasoline is potential energy.
• The energy in a moving car is kinetic energy.

When a body possesses energy, it can perform work.

WORK AND KINETIC ENERGY

The work done on a body by a force is directly related to the resulting change in the body’s motion.

If the object starts moving from rest

If the object starts moving


SAMPLE PROBLEM

1. If a 1,500kg car is moving at the rate of 20m/s, what is its kinetic energy in Joules?
2. A bullet, of mass 200g, is fired with a speed of 400m/s. If it stopped in 100meters, find (a) its KE and (b) the force needed to stop it.
3. Find the velocity of a ball with a mass of 1 kg after it has acquired 25 joules of kinetic energy.
4. From the Figure below, we found that the total work done by the forces was 500N.m = 500 J. Hence, the kinetic energy of the box must increase by 500 J. suppose the initial speed V₁ is 4m/s and the mass of the box is 10kg. What is the final speed?



GRAVITATIONAL POTENTIAL ENERGY

When a gravitational force acts on a body while the body undergoes a vertical displacement, the force does work on the body. This work can be expressed conveniently in terms of the initial and final position of the body.

When moving upward (height) or vertically from the starting point it is positive but when the body moves downward it is negative

The direction of w is opposite to the upward displacement, and the work done by this force is

Gravitational potential energy, U = the product of the weight mg and the height Y above the reference level (the origin of coordinates).

When the body moves downward, Y decreases, the gravitational force does positive work, and the potential energy decreases. When the body moves upward, the work done by the gravitational force is negative and the potential energy increases.

The gravitational potential energy of a body depends on its position.



The total work done by all forces is then
TOTAL MECHANICAL ENERGY = the sum of kinetic and potential energies.


If W _other is equal to zero:

Sample Problem:

1. A man holds a ball of mass m = 0.2kg at rest in his hand. He then throws the ball vertically upward. In this process, his hand moves up 0.5m before the ball leaves his hand with an upward velocity of 20m/s. Discuss the motion of the ball from the work-energy standpoint, assuming g = 10m/s²

2. Suppose a child of mass 25.0kg slides down a slither of radius R = 3.00m, but his speed at the bottom is only 3.00 m/s. What work was done by the frictional force acting on the child?
For the same initial speed, the speed is the same at all points at the same elevation.

=quantity is the general expression for the gravitational potential energy of the body attracted by the earth.

If the gravitational force is the only force on the body, then W_grav is equal to the change in kinetic energy from r₁ to r₂

The total mechanical energy of the body, the sum of its kinetic energy and potential energy

3. In Angelbert‘s story “Earthlings to the Moon” (written in 2005) three men were shot to the moon in a shell fired from a giant cannon sunk in the earth in Florida. What muzzle velocity would be needed (a.) to raise a total mass m to a height above the earth equal to the earth’s radius; (b.) to escape from the earth completely? To simplify the calculation, neglect the gravitational pull of the moon.

Force in Circular Motion

FORCE IN CIRCULAR MOTION

Uniform Circular Motion refers to a motion along a circular path with a uniform speed but constantly changing direction. It is accelerated motion.

Frequency (f) = Refers to the number of complete revolutions per unit of time.

Centripetal acceleration (central acceleration) = it is primarily acts at right angles to its velocity.

Central Force = refers to the force that compels a body to move toward the center.

* If the mass is doubled, the force is also doubled, but doubling the speed will increase the force four times. If the radius is doubled, this will decrease the force twice as much.

Period (τ) is define as the time for one revolution and often stated as the reciprocal of frequency (1 / f).

Angular velocity refers to the rate of angular displacement. The rate of angular turned through.

* the direction of the velocity is always tangent to the circular path.

* radian as the unit of angle

Linear velocity is the product of angular velocity and its radius. V = r ω

Sample Problem:

1. If the radius of the circular path of a stone is 0.5m and its period is 0.5 sec, what is its constant speed?

2. What is the angular and linear velocity of a stone which makes 10 rev in 5 sec? The radius of the circular path is 0.5 m. First find the angle of turned through

3. Half kilogram mass is whirled in a horizontal circle of radius 2m. If it makes 5 revolutions in 5 seconds, find its (a.) constant speed, (b.) central acceleration and (c.) central force.

4. Find the central force required by a Hammer car ( mass, 3000kg) that round a curve of radius 35m at a speed of 8 m/s.

5. A mass of 2 kg is moving in a horizontal circle of 1m radius with an angular velocity of 3 rad/s. What is the required central force?

6. A 50 gram body fastened to one end of a string 0.75m long, is made to move with uniform circular motion at 3 revolutions per second on a smooth horizontal surface, with one end of the string as its center, find (a.) the speed of the body and (b.) the central force

7. A small plastic box mass 0.200kg revolves uniformly in a circle on a frictionless horizontal surface such as an air hockey table. It is attached by a cord 0.200m long to a pin se in he surface. If the body makes two complete revolution per second, find the force F exerted on it by the cord.