Worked Examples-Calculation Part 1 - Work, Energy and Power

Considered Formula;

$$\text {i)}\quad W.d = F \times d$$ $$\qquad \text {Where}~~W.d \to Work~done,\quad F \to Force \\\qquad d \to Distance~moved$$ $$\text {ii)}\quad K.E = \frac {1}{2}mv^2$$ $$\qquad \text {Where}~~K.E \to Kinetic~energy,\quad m \to Mass \\\qquad v \to Velocity$$ $$\text {iii)}\quad P.E = mgh$$ $$\qquad \text {Where}~~P.E \to Potential~energy,\quad m \to Mass \\\qquad g \to Acceleration~due~to~gravity, \quad h \to height$$ $$\text {iv)}\quad P = \frac {W.d}{t}$$ $$\qquad \text {Where}~~P \to Power,\quad W.d \to Work~done \qquad t \to Time$$ $$\text {v)}\quad P = \frac {E}{t}$$ $$\qquad \text {Where}~~P \to Power,\quad E \to Energy \qquad t \to Time$$

Example 1

A man moves a stone by a force of 45 N to distance of 4 m long. What is the work done by the man?

Solution

Data given

Force (F) =45 N

Distance (d) = 4 m

Work done (W.d) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~W.d = F \times d$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 45 \times 4$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 180~Joules$$

The work done by the man = 180 Joules

Example 2

Abu Nufayl lifts a sack of rice by a force of 250 N against gravity to 2 m high. Calculate his work done

Solution

Data given

Force (F) =250 N

Distance (d) = 2 m

Work done (W.d) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~W.d = F \times d$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 250 \times 2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 500~Joules$$

The work done by Abu Nufayl = 500 Joules

Example 3

A car of mass 1000 Kg moves at a velocity of 5 m/s. calculate the kinetic energy possessed by the car.

Solution

Data given

Mass (m) =1000 Kg

Velocity (d) = 5 m/s

Kinetic energy (K.E) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~K.E = \frac {1}{2} mv^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {1}{2} \times 1000 \times 5^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {1}{2} \times 1000 \times 25$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 500 \times 25$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 12500~Joules$$

The kinetic energy possessed by the car = 12500 Joules

Example 4

Find the kinetic energy of the glass marble of mass 24 g travelling at a speed of 40 m/s.

Solution

Data given

$$\text {Mass (m)}= 24 g = \Biggl( \frac {24}{1000}\Biggr)~Kg = 0.024 Kg$$

Velocity (d) = 40 m/s

Kinetic energy (K.E) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~K.E = \frac {1}{2} mv^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {1}{2} \times 0.024 \times 40^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {1}{2} \times 0.024 \times 1600$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 0.012 \times 1600$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 19.2~Joules$$

The kinetic energy possessed by the car = 19.2 Joules

Example 5

A parcel of mass 36 Kg rests at the top of building 6 m high. Calculate the Potential energy possessed by The parcel.
(Use acceleration due to gravity = 10 m/s²) .

Solution

Data given

Mass (m) = 36 Kg

Height (h) = 6 m

Acceleration due to gravity (g) = 10 m/s²

Potential energy (P.E) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~P.E = mgh$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 36 \times \times 10 \times 6gh$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 2160~Joules$$

The Potential energy possessed by The parcel = 2160 Joules

Example 6

A mango of mass 50 g falls from tree 4 m high. If it moves at velocity of 80 m/s just before reaching the ground, calculate Potential energy possessed by the mango before falling
(Use acceleration due to gravity = 10 m/s²) .

Solution

Data given

$$\text {Mass (m)}= 50 g = \Biggl( \frac {50}{1000}\Biggr)~Kg = 0.05 Kg$$

Height (h) = 4 m

Acceleration due to gravity (g) = 10 m/s²

Potential energy (P.E) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~P.E = mgh$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 0.05 \times \times 10 \times 4gh$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 2~Joules$$

The Potential energy possessed by The parcel = 2 Joules

Example 7

A man moves a stone by a force of 45 N to distance of 4 m long in 3 seconds. Calculate power of the man?

Solution

Data given

Force (F) = 45 m

Distance (d) = 4 m

Time (t) = 3 s

Power (P) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~P = \frac {W.d}{t}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {F \times d}{t}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {45 \times 4}{3}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 60~Watt$$

The power of the man = 60 Watt

Example 8

How much power is required to accelerate a 1000 Kg car from rest to 6 m/s in 8 s?

Solution

Data given

Mass (m) = 1000 Kg

Velocity (v) = 6 m/s

Time (t) = 8 s

Power (P) =?

First find the Kinetic energy

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~K.E = \frac {1}{2} mv^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {1}{2} \times 1000 \times 6^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 500 \times 36$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 18000~Joules$$

From;

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~P = \frac {E}{t}$$

Hence;

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~P = \frac {K.E}{t}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \frac {18000}{6}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 2250~Watt$$

The power required to accelerate the car = 2250 Watt