Elementary ideas of work, power, and energy Questions

Correct Answer

📖 Governing Law / Core Concept

The work done \( W \) by a force is given by the formula: where \( F \) is the force applied, \( d \) is the displacement, and \( \theta \) is the angle between the force and the direction of displacement.

🧪 Step-by-Step Breakdown

The force is applied anti-parallel to the displacement, therefore: \( \theta = 180^\circ \) Calculating the work done: Substituting \( \cos 180^\circ = -1 \): Thus, the work done is negative.

🔍 Option Analysis

- Option (a): Zero - Incorrect, as work is negative. - Option (b): Positive - Incorrect, as work is negative. - Option (c): Infinite - Incorrect, as work does not approach infinity in this scenario. - Option (d): Negative - Correct, as derived above.

⚡ Mnemonic / Speed-Run

Remember: "Work and force dance; when they clash, work is negative." Forces opposing displacement yield negative work.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The first law of thermodynamics states that:

Where: \(\Delta U\) = change in internal energy, Q = heat transferred, W = work done.

🧪 Step-by-Step Breakdown

Given that the work done (W = 0), the equation simplifies to:

This indicates that the change in internal energy is directly equal to the heat transferred in or out of the system.

🔍 Option Analysis

• Option (b): Change in internal energy of the system is less than heat transferred. Incorrect; they are equal.
• Option (c): Change in internal energy of the system is more than the heat flow. Incorrect; they are equal.
• Option (d): Cannot be determined. Incorrect; it is determinable.

⚡ Mnemonic / Speed-Run

Remember: Energy transfer equals heat flow when no work is done.

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The moment of inertia (\(I\)) about the center of mass for a solid disc and a solid sphere is defined as:

🧪 Step-by-Step Breakdown

Given that both the disc and sphere have the same mass (\(m\)) and radius (\(r\)), we can directly substitute the values into their respective moment of inertia formulas.

Now comparing the two:

This shows that the moment of inertia of the disc is greater than that of the sphere.

Correct Answer

📖 Governing Law / Core Concept

The momentum \( p \) of an object is defined as the product of its mass \( m \) and its velocity \( v \):

🧪 Step-by-Step Breakdown

Upon bouncing, the direction of the velocity of the ball changes rapidly, resulting in:

Where \( v' \) is the velocity after the bounce (upward) and \( v \) is the velocity before (downward). This rapid change implies:

🔍 Option Analysis

• A: Its speed - While speed may change, it doesn't change suddenly due to the velocity change being instantaneous.
• C: Its kinetic energy - Kinetic energy also changes but not necessarily suddenly across all bounces.
• D: Its potential energy - This alters gradually during the ascent and descent of the ball.

⚡ Mnemonic / Speed-Run

Remember: Momentum is impacted by the velocity change; hence, think "Momentum suddenly changes when direction does!"

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The rotational kinetic energy of a body is given by the formula:

Where \( K.E. \) is kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.

🧪 Step-by-Step Breakdown

For a thin ring:

Thus, the kinetic energy becomes:

For a thin disc:

So, the kinetic energy for the disc is:

Comparing the two kinetic energies:

• K.E. of Ring: \( \frac{1}{2} M R^2 \omega^2 \)
• K.E. of Disc: \( \frac{1}{4} M R^2 \omega^2 \)

Since \( \frac{1}{2} > \frac{1}{4} \), the ring has higher kinetic energy than the disc.

🔍 Option Analysis

• Option (b): The disc has higher kinetic energy. Incorrect
• Option (c): The ring and the disc have the same kinetic energy. Incorrect
• Option (d): Kinetic energies of both bodies are zero since they are not in linear motion. Incorrect

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The work done \(W\) by a force is defined as: where \(F\) is the magnitude of the force, \(S\) is the displacement, and \(\theta\) is the angle between the force and displacement vectors.

🧪 Step-by-Step Breakdown

For negative work, the angle \(\theta\) must be in the range where \( \cos(\theta) < 0\). This occurs when: \[ \theta = 180^\circ \] Thus, the work done can be represented as: Here, \(W < 0\) confirms that negative work is done when force and displacement are in opposite directions (anti-parallel).

🔍 Option Analysis

- A: When perpendicular, \(W = 0\) since \(\cos(90^\circ) = 0\). No work. - B: When parallel, \(W = F \cdot S\), yielding positive work. - C: Correct; results in negative work as discussed. - D: Equal magnitudes alone do not determine work's sign.

⚡ Mnemonic / Speed-Run

Remember: "Anti-parallel means opposing, causing negative work in motion."

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The concepts involved here relate to thrust (\( T \)) and impulse (\( I \)), which are defined as follows: - Thrust, \( T = F \cdot \Delta t \) - Impulse, \( I = F \cdot \Delta t \) Where: - \( F \) = Force (measured in Newtons, \( N \)) - \( \Delta t \) = Time duration (measured in seconds, \( s \)) Thus, the ratio \( \frac{T}{I} \) can be expressed and analyzed.

🧪 Step-by-Step Breakdown

1. Since \( T \) and \( I \) are both derived from the same principles of force over time: \[ \frac{T}{I} = \frac{F \cdot \Delta t}{F \cdot \Delta t} = 1 \] Therefore, the ratio \( \frac{T}{I} \) is dimensionless. 2. The frequency (\( f \)) is defined as cycles per second: \[ f = \frac{1}{T} \] 3. Resultantly, we see that the dimensional formula for thrust and impulse leads us to understand that both can be analyzed in terms of time-based ratios, primarily relating to cyclical motion or frequency.

🔍 Option Analysis

- Option (b) Speed: Dimensions are \( [L][T^{-1}] \) - Option (c) Wavelength: Dimensions are \( [L] \) - Option (d) Acceleration: Dimensions are \( [L][T^{-2}] \) The only concept sharing dimensional properties with our derived ratio is frequency.

⚡ Mnemonic / Speed-Run

- Remember: Frequency is the "cycle count per second,” directly linking it to thrust and impulse's time-related mechanics.

6. Visual Suggestion

📖 Factual Verification & Reference

Solution

Option (a) is correct.

Explanation:

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

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