Stability and equilibrium of bodies Questions

Correct Answer

📖 Governing Law / Core Concept

Newton's second law of motion states: F = ma, where F is the net force acting on an object, m is its mass, and a is its acceleration.

🧪 Step-by-Step Breakdown

Given:

• Mass (\(m\)) = 2.0 kg
• Initial Velocity (\(u\)) = 10 m/s
• Final Velocity (\(v\)) = 0 m/s
• Distance (\(s\)) = 20 m

Using the third equation of motion:

Substituting the known values:

Simplifying gives:

Now applying Newton's second law:

The negative sign indicates that the frictional force acts in the direction opposite to the motion.

🔍 Option Analysis

Option (b) 20 N, (c) 40 N, and (d) 50 N are not possible as they do not satisfy the equations derived.

⚡ Mnemonic / Speed-Run

To remember laws of motion: "Force equals mass times acceleration."

📊 Visual Diagram Suggestion:

Correct Answer

📖 Governing Law / Core Concept

The principle relevant here is the impulse-momentum theorem, which states:

Impulse is the product of the average force and the time duration over which the force acts.

🧪 Step-by-Step Breakdown

When a fielder catches a cricket ball, he effectively receives momentum from the ball, which is equal to:

where \( m \) is the mass of the ball (in kg) and \( v \) is its velocity (in m/s).

By pulling his hands backward, the fielder increases the time (\( \Delta t \)) over which the ball decelerates to rest. This longer time reduces the average force (\( F \)) experienced, calculated as:

Rearranging gives:

This indicates that increasing \( \Delta t \) (catching time) reduces force \( F \), mitigating injury and improving catching success.

🔍 Option Analysis

• Option B: Incorrect, as increasing the acceleration is undesirable when catching a ball.
• Option C: Incorrect, as velocity is not increased; the goal is to decelerate the ball.
• Option D: Incorrect, as the above explanations suffice.

⚡ Mnemonic / Speed-Run

Remember: More Time = Less Force when catching. This mnemonic highlights the benefit of time during collisions.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

According to the Law of Conservation of Energy, energy cannot be created or destroyed in an isolated system.

🧪 Step-by-Step Breakdown

In an isolated system, no matter or energy enters or leaves the system. Therefore, the total energy remains constant.

For a non-isolated system, external forces can do work, resulting in energy being transformed or transferred:

• Isolated System: \( E_{total} = E_{initial} \)
• Non-Isolated System: \( E_{total} = E_{initial} + W_{ext} \) (where \( W_{ext} \) is work done by external forces)

🔍 Option Analysis

• B: Non-isolated systems experience energy transfer.
• C: Both cannot guarantee energy conservation.
• D: None of the above contradicts conservation laws.

⚡ Mnemonic / Speed-Run

Remember: "Isolated = Conserved, Non-isolated = Transferred."

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The principle governing the motion of the ball is the Conservation of Mechanical Energy, which states that the total mechanical energy (sum of kinetic energy (KE) and potential energy (PE)) in a closed system remains constant if only conservative forces, like gravity, act on the body.

🧪 Step-by-Step Breakdown

As the ball is thrown upwards from point A:

• At point A: Kinetic Energy is maximum, Potential Energy is minimum.
• At point B (highest point): Velocity is zero, thus Kinetic Energy is zero, and Potential Energy is maximum.

By Conservation of Energy:

Since \( PE_A = 0 \) (taking point A as reference), and \( KE_B = 0 \) (at highest point B), we have:

Thus, at point B, the entire kinetic energy at A (when the ball was thrown) has been converted into potential energy.

🔍 Option Analysis

• Option (b): Incorrect. Kinetic Energy at A is not equal to Potential Energy at A, as KE is maximum and PE is minimum there.
• Option (c): Incorrect. At point B, KE is zero, whereas PE is maximum.
• Option (d): Incorrect. At point B, KE is zero while at A it is maximum.

⚡ Mnemonic / Speed-Run

Remember: "Up is Potential, Down is Kinetic." Energy shifts from kinetic to potential as height increases.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

According to Newton's laws of motion, weight \( W \) is the gravitational force acting on an object, related to its mass \( m \) by the equation:

Where:

• \( W \) = Weight (N)
• \( m \) = Mass (kg)
• \( g \) = Acceleration due to gravity (m/s²)

🧪 Step-by-Step Breakdown

Mass is defined as the measure of the amount of matter in an object. It remains constant regardless of location, thereby acting as a constant of proportionality in the weight equation. Weight, on the other hand, is variable - it depends on gravitational force, which changes with the distance from the center of the Earth and the local gravitational field strength.

To illustrate:

• If weight changes, mass remains the same.
• This confirms that mass is a constant of proportionality, while weight varies.

🔍 Option Analysis

Option (a) incorrectly states that weight is a constant of proportionality. Option (c) is false since mass is indeed a constant of proportionality. Option (d) mistakenly labels weight as a universal constant; however, it varies with gravitational force.

⚡ Mnemonic / Speed-Run

Remember: "Mass is Matter's Measure, Weight Wavers with Gravity."

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

Work (\( W \)) done by a constant force (\( F \)) is given by the formula: where: - \( W \) = Work (in Joules) - \( F \) = Force (in Newtons) - \( d \) = Displacement (in meters) - \( \theta \) = Angle between force and displacement direction In this case, \(\theta = 0^\circ\) (force in the same direction as displacement).

🧪 Step-by-Step Breakdown

To find the work done: - Given: \( F = 4 \, \text{N} \), \( d = 25 \, \text{cm} = 0.25 \, \text{m} \) Work calculation:

🔍 Option Analysis

- **Option (b)**: \( 2 \, \text{N} \cdot 1 \, \text{m} = 2 \, \text{J} \) (not correct). - **Option (c)**: \( 1 \, \text{N} \cdot 0.01 \, \text{m} = 0.01 \, \text{J} \) (not correct). - **Option (d)**: \( 1 \, \text{N} \cdot 0.5 \, \text{m} = 0.5 \, \text{J} \) (not correct).

⚡ Mnemonic / Speed-Run

Remember: \( 1 \, \text{J} = 1 \, \text{N} \cdot 1 \, \text{m} \). Keep units consistent when calculating work; convert to meters when necessary.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The concept involved in this scenario is based on Newton's First Law of Motion, which states that an object in motion will remain in motion unless acted upon by an external force. Circular motion implies a constant speed but a continuously changing velocity due to the change in direction.

🧪 Step-by-Step Breakdown

A car moving in uniform circular motion maintains a constant speed; however, its direction is constantly changing. The key factors involved include:

• Velocity (\( v \)): Change in both magnitude and direction.
• Acceleration (\( a \)): Centripetal acceleration required to keep the car moving in a circle, given by:
  \[ a = \frac{v^2}{r} \]
  where \( r \) is the radius of the circular path.
• Momentum (\( p \)): Defined as \( p = mv \), where \( m \) is the mass of the car. Since the direction of the velocity vector changes, the momentum changes continuously as well.

🔍 Option Analysis

Option (a): Incorrect; speed remains constant, but direction changes.
Option (c): Incorrect; a change does occur in both direction and speed.
Option (d): Incorrect; momentum cannot remain constant if velocity is changing.

⚡ Mnemonic / Speed-Run

Remember: "Centripetal means center-seeking," which reminds that centripetal force is what keeps an object in a circular path, thereby influencing its velocity and momentum even when speed is uniform.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The relevant formula governing the motion of an upward-moving object is derived from the equations of motion under uniform acceleration due to gravity. Specifically, the maximum height \(h\) attained can be given by: \[ h = \frac{v^2}{2g} \] where \(v\) is the initial velocity and \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)).

🧪 Step-by-Step Breakdown

Given that the maximum height \(h = 20 \, \text{m}\): \[ 20 = \frac{v^2}{2 \times 9.81} \] Rearranging gives: \[ v^2 = 20 \times 2 \times 9.81 = 392.4 \] Taking the square root: \[ v = \sqrt{392.4} \approx 19.8 \, \text{m/s} \] Thus, \(v\) is approximately \(20 \, \text{m/s}\).

🔍 Option Analysis

Options A (8 m/s), B (12 m/s), and C (16 m/s) are incorrect as they would not provide enough initial velocity to reach the maximum height of 20 m given the gravitational acceleration.

⚡ Mnemonic / Speed-Run

Remember \(h = \frac{v^2}{2g}\) as "Height equals velocity squared over twice gravity." This formula summarizes the energy conversion from kinetic to potential energy.

6. Visual Suggestion

📖 Factual Verification & Reference

Solution

Option (a) is correct.

Explanation:

Correct Answer

📖 Governing Law / Core Concept

The work done \( W \) by a force \( F \) acting on an object and causing a displacement \( s \) is given by the formula:

Here, \( \theta \) is the angle between the force and displacement vectors.

🧪 Step-by-Step Breakdown

If the displacement is perpendicular to the force:

Thus, substituting into the work formula:

Therefore, the work done is zero when the force acts perpendicular to the displacement.

🔍 Option Analysis

• Option A: Incorrect. Displacement in the opposite direction of the force yields negative work.
• Option B: Incorrect. Displacement in the same direction yields positive work.
• Option D: Incorrect. Valid scenarios exist that contradict this.

⚡ Mnemonic / Speed-Run

Remember: Perpendicular = Zero Work. This can be visualized as vectors at right angles not contributing to the work done.

6. Visual Suggestion

📖 Factual Verification & Reference