Problem 379 - JEE Physics

Question

A lift of mass \(1000 \mathrm{~kg}\) is moving with an acceleration of \(1 \mathrm{~ms}^{-2}\) in upward direction Tension developed in the rope of lift is \(\mathrm{N}\left(\mathrm{g}=9.8 \mathrm{~ms}^{-2}\right)\) (A) 9800 (B) 10,000 (C) 10,800 (D) 11,000

Step-by-Step Solution

Answer

The tension developed in the rope of the lift is \(10,800 N\).

Step 1: Identify the given values

We are given: - Mass of the lift (m) = 1000 kg - Acceleration due to gravity (g) = 9.8 m/s² - Acceleration in the upward direction (a) = 1 m/s²

Step 2: Use Newton's second law of motion to calculate the tension

We will use Newton's second law of motion to calculate the tension (T) in the rope: Tension (T) = Mass (m) × (g + acceleration) Substitute the given values into the equation: T = 1000 kg × (9.8 m/s² + 1 m/s²)

Step 3: Solve for the tension

Now, solve for the tension: T = 1000 kg × (10.8 m/s²) T = 10,800 N

Step 4: Identify the correct option

From our calculation, we have found that the tension developed in the rope of the lift is 10,800 N. Comparing this value to the given options, we can see that the correct answer is: (C) 10,800

Key Concepts

Understanding Tension Calculation

Calculating tension in a rope involves understanding the forces acting on the object it supports. Tension can be thought of as the force transmitted through a string, cable, or in this case, a rope when it is pulled tight by forces acting from opposite ends. In the original exercise, the lift's motion involves both its weight and the additional upward acceleration.

To determine the tension in the rope, we utilize Newton's Second Law of Motion, which is a fundamental principle in mechanics. According to this law, the force required to move an object is equal to the mass of the object multiplied by its acceleration:

• Weight force acting downwards = mass (m) x gravitational acceleration (g)
• Force required to accelerate the lift upwards = mass (m) x additional acceleration (a)

Thus, the total tension force (T) in the rope can be found by combining these two forces:\[ T = m (g + a) \]This is because the tension must support both the lift's weight and its upward acceleration. The calculated tension of 10,800 N indicates how much force the rope needs to exert to move the lift as described.

Exploring Acceleration in Motion

Acceleration is a key component in understanding motion. In physics, acceleration refers to the rate of change of velocity of an object. It's a vector quantity, meaning it has both magnitude and direction. In our scenario, the lift accelerates upwards at 1 m/s². This affects how we calculate forces acting on the lift, including the tension in its cable.

Here are some vital points to remember about acceleration:

• Acceleration can result from a change in speed, direction, or both.
• When an object is in free fall, its acceleration is due to gravity, typically 9.8 m/s² towards the Earth's center.
• In our lift problem, the total upward force required is a combination of fighting gravity and adding the desired acceleration.

The lifting mechanism creates this additional acceleration which contributes to the increased force needed to move the lift upwards, further illustrating how acceleration is manipulated to achieve desired motion.

Force and Motion Interaction

The interplay of force and motion is crucial in understanding physics concepts like tension and acceleration. Force is any interaction that tends to change the motion of an object. Newton's Laws of Motion provide the foundation for understanding these interactions. Particularly, Newton's Second Law relates force, mass, and acceleration through the equation:\[ F = m imes a \]In this lift scenario, we deal with forces acting vertically, such as gravity pulling it downwards and the tension in the rope pulling upwards.

Key aspects to consider are:

• Newton's First Law states that an object at rest will stay at rest unless acted on by a force. Similarly, an object in motion will stay in motion at constant velocity if there's no net force.
• Newton's Second Law helps us calculate the exact forces needed to achieve desired acceleration, as seen in our example where tension in the lift's rope counteracts gravitational force and adds extra force to cause upward motion.
• Newton’s Third Law, "for every action, there is an equal and opposite reaction," is evident in the tension exerted by the rope as it pulls the lift upwards.

Understanding these interactions explains how accelerated motion is achieved in real-world scenarios, such as our lift problem, and enhances our grasp of fundamental physics principles.