WORK, ENERGY AND POWER

The Ninja, a roller coaster at Six Flags over Georgia, has a height of 122 ft and a speed of 52 Mph. The potential energy due to its height changes into kinetic energy of motion.

WORK

Work is done by force when there is a force applied on the body and the body must move with a displacement in line with the force applied.�⃑� �⃑� �⃑�

𝜃 𝜃 𝜃

∆ 𝑠�⃑�𝜃 ∆ 𝑠𝐹 ¿∨¿¿

= angle bet. and

= component of parallel with

𝑊=𝐹 ¿∨¿∆ 𝑠=𝐹 ∆𝑠 cos𝜃 ¿Work done by constant force

𝑾 canbe+ 𝑖𝑓𝑓 𝐹 ∥∆ 𝑠(0𝑜 ≤𝜃<90𝑜)

−𝑖𝑓𝑓 𝐹 ∥∆ 𝑠(90𝑜<𝜃 ≤ 180𝑜)𝟎 𝑖𝑓𝑓 𝐹 ⊥∆𝑠 (𝜃=90𝑜)

Units of work:

joule, J (1 J = 1 N-m)

erg (1 erg = 1 dyne-cm)

ft-lb

Example 01

Demi horizontally pushes the 200-N crate in a rough horizontal plane with a constant force of 90 N to continuously move it in uniform motion at a distance of 100 m. What is the total work done on the crate?

Energy is anything that can be converted into work; i.e., anything that can exert a force through a distance.

Energy is the capability for doing work.

ENERGY

Unit of energy is the same to the unit of work.

Other units used:

calorie

British Thermal Unit (Btu)

kilowatt-hour

Kinds of Mechanical Energy

1. Kinetic Energy, K – “speed”

2. Potential Energy, U – “position” or “condition”

a. Gravitational PE, Ug

b. Elastic PE

c. Electric PE

Transit Energies: KE and Heat

”

Work done and Kinetic Energy𝑣𝑜 𝑣

𝑃 𝑃∆ 𝑠

𝐹=𝑚𝑎 𝑎=𝑣2 −𝑣𝑜❑

2

2∆ 𝑠𝐹

¿∨¿=𝑚(𝑣2 − 𝑣𝑜❑2

2 ∆ 𝑠 )¿𝐹

¿∨¿∆ 𝑠=12𝑚𝑣2 − 1

2𝑚𝑣𝑜

2¿ 𝐾=12

𝑚𝑣2Kinetic energy

Work-Energy Theorem

Work done on the body by resultant forces is its change in kinetic energy

WORK DONE BY GRAVITY (WEIGHT) AND GRAVITATIONAL POTENTIAL

ENERGY

𝑊𝑤=𝑤 ∆ 𝑠cos 𝜃

𝑊𝑤=𝑚𝑔 ( 𝑦− 𝑦𝑜 ) cos180

𝑊𝑤=𝑚𝑔 𝑦 𝑜−𝑚𝑔𝑦

𝑈𝑔=𝑚𝑔𝑦 Gravitational potential energy

Work done on the body is its negative change in potential energy

𝑤

𝑤

∆ 𝑠

𝑦

𝑦 𝑜

REVIEW FIRST:

𝑊=𝐹 ¿∨¿∆ 𝑠=𝐹 ∆𝑠 cos𝜃 ¿ Work

𝐾=12

𝑚𝑣2 Kinetic Energy

𝑈𝑔=𝑚𝑔𝑦 Gravitational Potential Energy

𝑊=∆𝐾 Work-Energy Theorem

since𝑊=∆𝐾

𝑊 1=− ∆𝑈

∴− ∆𝑈+𝑊 h𝑜𝑡 𝑒𝑟=∆ 𝐾

𝑈𝑜 −𝑈+𝑊 h𝑜𝑡 𝑒𝑟=𝐾 − 𝐾𝑜

𝐾 𝑜+𝑈𝑜+𝑊 h𝑜𝑡 𝑒𝑟=𝐾 +𝑈 Law of Conservation of Energy

Initial energy = final energy

𝑊 2+…=𝑊 h𝑜𝑡 𝑒𝑟=work done by other forces

𝑊 1+𝑊 2+…=∆ 𝐾

Examples: Use energy methods to solve all problems

1. A bus slams on brakes to avoid an accident. The thread marks of the tires is 25 m long. If , what was the speed of the bus before applying brakes?

2. A 1.50-kg book is dropped from a height of 15.0 m from the ground. Find its potential and kinetic energy when it is 6.0 m from the ground.

3. A small rock with a mass of 0.20kg is released from rest at point A, which is at the top edge of a large hemispherical bowl with radius R = 0.80m. Assume that the size of the rock is small in comparison to the radius of the bowl, so the rock can be treated as particle, the work done by the friction when it moves from point A to point B at the bottom of the bowl is -0.22J. What is the speed of the rock when it reaches point B?

POWER

Power is defined as the rate at which work is done.

𝑷=∆𝑾∆ 𝒕 Power

Units of Power:

watt, W

erg/s

foot=pound per second (ft-lb/s)

horsepower

POWER AND VELOCITY

Recall average speed or constant velocity:

So that

Since and

𝑃=𝐹𝑑𝑡

=𝐹 𝑣𝑡

𝑡

𝑃=𝐹 𝑣 Power at constant velocity

EXAMPLE OF POWER

P = 2200 W=2.2 kWP = 2200 W=2.2 kW

What power is consumed in lifting a 70.0-kg robber 1.6 m

in 0.50 s?𝑃=

𝑊∆𝑡

𝑃=𝐿𝑦𝑡 𝑃=

𝑚𝑔 𝑦𝑡

𝑃=(70.0 kg)(9.8 m / s2)(1.6 m)

0.50 s

MORE PROBLEMS

1. Tarzan swings on a 30.0-m-long vine initially inclined at an angle of 37.0o with the vertical. What is his speed at the bottom of the swing (a) if he starts from rest? (b) if he pushes off with a speed of 4.00m/s? hint: the work done by tension is zero.

2. A 45.0-kg block of wood initially at rest is pulled by a cord from the bottom of a 27.0o inclined plane. The tension of the cord is 310 N parallel to the plane. After travelling a distance of 2.0 m , the speed of the block is 5.0 m/s. (a) what is the work done by friction? (b) what is the coefficient of friction?

3. A 750-N box is pulled in a rough horizontal plane by a motor driven cable. The coefficient of kinetic friction between the box and the plane is 0.40. (a) How much work is required to pull it 60 m at a constant speed of 2.0 m/s? (b) What power must the motor have to perform this task?

Use energy methods to solve all problems