Question

\(1 \mathrm{~kg}\) apple gives \(25 \mathrm{KJ}\) energy to a monkey. How much height he can climb by using this energy if his efficiency is \(40 \%\). (mass of monkey \(=25 \mathrm{~kg}\) and \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (A) \(20 \mathrm{~m}\) (B) \(4 \mathrm{~m}\) (C) \(30 \mathrm{~m}\) (D) \(40 \mathrm{~m}\)

Short answer

The monkey can climb 40 meters with the energy provided by the 1 kg apple, so the correct answer is (D) \(40 \mathrm{~m}\).

Step-by-step solution

Convert the given energy to work done by the monkey

The monkey has an efficiency of 40%, which means only 40% of the energy from the apple is converted into work done by the monkey. Therefore, we need to calculate the actual work done by the monkey: Work done = Efficiency × Energy from apple \( Work \, done = 0.40 \times 25 \, KJ \)

Convert work done to potential energy

The work done by the monkey is equal to the change in potential energy (PE) as it climbs. We can write the potential energy equation as: PE = Mass × Gravitational acceleration × Height In our case, the mass of the monkey is 25 kg, and gravitational acceleration is 10 m/s². We are interested in the height climbed, so we can rewrite the equation as: Height = PE / (Mass × Gravitational acceleration) Now let's calculate the height climbed by the monkey.

Calculate the height climbed

Using the values calculated in Step 1 and Step 2, we can now find the height climbed by the monkey: Height = \( \frac{Work \, done}{(Mass \, × \, Gravitational \, acceleration)} \) Height = \( \frac{0.40 \times 25 \, KJ}{(25 \, kg \, × \, 10 \, m/s^{2})} \) First, we need to convert KJ to J (1 KJ = 1000 J): Height = \( \frac{0.40 \times 25000 \, J}{(25 \, kg \, × \, 10 \, m/s^{2})} \) Now, we can calculate the height: Height = \( \frac{10000 \, J}{(250 \, kg \cdot m/s^{2})} \) Height = 40 m #Conclusion# The monkey can climb 40 meters with the energy provided by the 1 kg apple, so the correct answer is (D) \(40 \mathrm{~m}\).

Key concepts

Work-Energy Principle

The Work-Energy Principle forms the foundational framework for understanding how energy is transferred and transformed in processes that involve motion. In essence, it explains how the amount of work done on an object corresponds to its change in energy. Work done is defined as the force applied to an object multiplied by the distance over which the force acts.

For the monkey scenario, only 40% of the energy the monkey gains from the apple is used to perform work because of its 40% efficiency. Efficiency is a measure of how well the energy input is converted to useful work.

• Calculate Work Done: To find the actual work done by the monkey, we use the equation: \( \text{Work done (J)} = \text{Efficiency} \times \text{Energy from apple (J)} \)

Once the work done is computed, it becomes equal to the potential energy gained by climbing, because the work-energy principle states that the work done results in a change of the energy state of the system. In this case, the system is the monkey climbing upward.

Potential Energy

Potential energy is the energy stored in an object due to its position relative to a force, in this case, the gravitational force of Earth. When the monkey climbs, it gains potential energy because it is moving against gravity to a higher position.

The formula for gravitational potential energy is given by:

• \( \text{Potential Energy (PE)} = \text{Mass} \times \text{Gravitational acceleration} \times \text{Height} \)

For the monkey, knowing the mass (25 kg), gravitational acceleration (10 m/s²), and solving for height using the work converted into potential energy, allows us to find how high it can climb. Simply rearrange the formula to solve for height:

• \( \text{Height} = \frac{\text{Work done}}{\text{Mass} \times \text{Gravitational acceleration}} \)

In this calculation, all energy from work is transformed into potential energy as no other forces, like friction, oppose the climb.

Gravitational Acceleration

Gravitational acceleration is the rate at which an object accelerates due to the force of gravity when falling towards Earth without any resistance. Its standard value is approximately 9.81 m/s² on Earth's surface but is approximated to 10 m/s² for easier calculations, as used in most textbook problems and in this exercise.

Gravitational acceleration is crucial, as it directly affects the amount of potential energy an object gains as it climbs upward or the speed it gains when falling downward.

• Influence on Potential Energy: Since potential energy depends on the gravitational acceleration \( (PE = m \times g \times h) \), any object gaining height, like the monkey climbing, gains energy proportional to this constant.
• Unit Understanding: Remember that gravitational acceleration has units of \( m/s^2 \), ensuring that calculations of energy and dynamics are in consistent units, like Joules for energy.

Through this parameter, we can accurately predict the behavior of masses in a gravitational field, applying the concept across physics to understand motion and energy transformations.