Lcr Circuit

Where time constant

The relationship between the radius of a nucleus and its mass number is given by the empirical formula: where is a constant. Taking the natural logarithm of both sides, we get: This shows that the graph of versus is a straight line with a slope of . Thus, the correct option is the one that shows a straight line.

Step 1: The wave velocity is given by:
From the given wave equation, we identify:

Step 2: The wave velocity is:

Step 3: Using , solving for :

To determine the number of lamps that can be lit, we first calculate the available power from the falling water.
Step 1: Given Data
- Height of the dam: m
- Mass flow rate of water: kg per hour
- Efficiency of conversion:
- Acceleration due to gravity: m/s²
- Power of each lamp: W
Step 2: Calculate Gravitational Potential Energy Per Second
The gravitational potential energy released per second is:
Since mass flow rate is given per hour, we convert it to per second:
Thus,
Step 3: Calculate Electrical Power Output
Only of this energy is converted into electrical energy:
Step 4: Calculate Number of Lamps
Each lamp requires W. The number of lamps that can be powered is:
Thus, the correct answer is:

Given: Radius of the ball: Density of the ball: Distance fallen: Viscosity of water: Acceleration due to gravity: We need to find the value of . Since the velocity of the ball does not change upon entering the water, the drag force and the gravitational force must be in equilibrium when the ball moves through the water at a constant velocity. The drag force acting on the ball in water is given by Stokes' Law: where is the velocity of the ball just before entering the water. The gravitational force acting on the ball is given by: For the ball to move with a constant velocity in water, the net force acting on it should be zero: Substituting the expressions for and : Simplifying this equation, we get: The ball falls freely under gravity through a distance before entering the water. The velocity of the ball just before entering the water is given by: Equating the two expressions for : Squaring both sides to solve for : Now, we need to consider the relative density, and thus we substitute with : So, the detailed solution confirms that the correct answer is:

Step 1: State the given data. The initial pressure inside the tyre, , is 4 times the atmospheric pressure : The initial temperature, , is given as: Step 2: Apply the adiabatic process for an ideal gas. For an adiabatic process, the relation between pressure and temperature is given by: where is the final pressure and is the final temperature. Since the tyre bursts, the final pressure will be equal to the atmospheric pressure : Step 3: Substitute the known values into the adiabatic equation. Substitute , , and into the equation: Step 4: Solve for the final temperature . Divide both sides by : Raise both sides to the power of to solve for : Step 5: Simplify the expression. Using the property of exponents, we can simplify the expression: Thus, the final temperature is:

The ozone layer plays a crucial role in protecting life on Earth by absorbing harmful radiation from the Sun. Let’s analyze the absorption properties of the ozone layer.
Step 1: Understanding the Ozone Layer
- The ozone layer is present in the stratosphere, approximately 10–50 km above the Earth’s surface.
- It is primarily composed of ozone molecules ( ), which absorb high-energy electromagnetic radiation.
Step 2: Absorption of Electromagnetic Waves
- The ozone layer does not absorb radio waves, which have very long wavelengths and low energy. Hence, option (A) is incorrect.
- Visible light passes through the ozone layer without significant absorption, allowing sunlight to reach Earth. Thus, option (B) is incorrect.
- Gamma rays ( -rays) are absorbed primarily by the Earth's atmosphere, but not specifically by the ozone layer. Hence, option (C) is incorrect.
- X-rays and ultraviolet (UV) rays are absorbed effectively by the ozone layer, preventing harmful radiation from reaching Earth's surface. This makes option (D) correct.
Step 3: Explanation of Correct Answer
- The ultraviolet radiation (UV-C and most of UV-B) from the Sun is absorbed by ozone, preventing damage to living organisms.
- X-rays from the Sun and cosmic sources are also absorbed by the ozone layer, preventing them from reaching the surface.
Thus, the correct answer is:

To determine the percentage rectifier efficiency of a half-wave rectifier, we use the formula:
where:
- is the DC power delivered to the load.
- is the AC power supplied to the rectifier.
Step 1: Given Data
- Peak voltage of AC supply:
- Frequency: Hz (not required for efficiency calculation)
- Load resistance:
- Forward resistance of diode:
Step 2: Calculate the RMS Value of AC Input Voltage
The RMS value of the input AC voltage is given by:
Substituting the values:
Step 3: Calculate the DC Output Voltage
For a half-wave rectifier, the DC output voltage is given by:
Since is unknown, we approximate as:
Step 4: Calculate the DC Power Delivered to Load
The DC output current is:
Thus, the DC power delivered to the load is:
Step 5: Calculate the AC Power Supplied to Rectifier
The AC power supplied to the rectifier is given by:
where is the equivalent resistance:
Step 6: Calculate Efficiency
Thus, the percentage rectifier efficiency is: