Problem 86 - Master Resource Book in JEE Main Physics

Question

One man takes 1 minute to raise a box to a height of \(1 \mathrm{~m}\) and another man takes \(\frac{1}{2}\) minute to do so. The energy of the two is (a) different (b) same (c) energy of the first is more (d) energy of the second is more

Step-by-Step Solution

Answer

(b) same

Step 1: Understand the Concept of Work and Energy

Energy in this context refers to the amount of work done to lift a box to a certain height. The work done (or energy) to lift the box is determined by the force applied (which is equal to the weight of the box) and the distance it is moved vertically, which is given by the formula: \[ W = m \cdot g \cdot h \]where \( m \) is the mass of the box, \( g \) is the acceleration due to gravity, and \( h \) is the height to which the box is lifted.

Step 2: Calculate Work Done by Both Men

Regardless of how long it takes to raise the box, the work done is the same for both men, as they both lift the box to a height of \(1\, \text{m} \) against gravity. Therefore, - Work done by the first man: \[ W_1 = m \cdot g \cdot 1 \]- Work done by the second man: \[ W_2 = m \cdot g \cdot 1 \]Since \( W_1 = W_2 \), the energy is the same for both.

Step 3: Conclusion Based on Work Done

Since both men perform the same amount of work in lifting the box, the energy used by both men is the same, irrespective of the time taken.

Key Concepts

Force and Motion

When discussing force and motion, we're diving into how objects start, stop, change speed, or direction. These concepts are key to understanding much of what we see in physics. It's important to grasp that force is a push or a pull on an object that can cause it to change its velocity. Motion, on the other hand, refers to the change in position of the object over time. When the first man lifts the box to a height of 1 meter, a force (the man's effort) acts against the gravitational pull. This force is necessary to move the box. When considering lifting objects, the motion - moving the box upward - is uniform because we assume no other forces like friction interfere. Newton's second law, which states that force equals mass times acceleration ( F = m imes a ), helps us understand that the applied force is crucial for initiating the motion upward against gravity. Understanding these principles helps in visualizing how the energy expended by both men relates directly to the force and motion involved.

Gravitational Force

Gravitational force is a fundamental concept in physics and describes the attractive force between two masses. Near the Earth's surface, this force gives weight to objects and acts downwards, toward the center of the Earth.In our exercise, the gravitational force is what both men must overcome to lift the box. This force is calculated by multiplying the mass of the box ( m ) by the gravitational acceleration ( g ), which is approximately 9.8 m/s² on Earth. The equation for gravitational force is: \[ F_g = m imes g \] When each man lifts the box to a height of 1 meter, they're doing work against this gravitational force. It's key to understand that the gravitational force remains constant; thus, the work done (which transfers energy) only depends on this force and the height the box is raised to.Comprehending gravitational force provides clarity on why the work done by both men is the same, even if their speed or force application differs. They both exert enough force to counteract gravity over the same distance.

Physics Problems

Physics problems often involve applying theoretical principles to practical situations. They require an analysis of various physical quantities, their interrelationships, and the formulation of equations that govern these metrics. In tackling physics problems like lifting a box, it is essential to:

• Identify all forces acting on the object, such as gravitational force.
• Understand the role of energy and work in the given context.
• Use appropriate formulas to determine quantities like force, work, or energy.

For this particular problem, the focus lies in understanding how time doesn't affect the energy or work done when both tasks involve lifting the same box to the same height. The energy is related to the work done, and since the work relies on the height and force applied (gravitational force), both outcomes are identical. By analyzing these physics problems step-wise, students can see first-hand how different concepts interconnect. This aids in grasping broader topics and solving more advanced physical challenges. It's also useful to note that while calculation methods may seem straightforward, the core understanding of concepts like work, energy, and gravitational effects provides the depth required for mastering physics.