A uniform rod of length L rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end. The rod is initially at rest. A bullet traveling parallel to the horizontal surface and perpendicular to the rod with speed v strikes the rod at its center and becomes embedded in it. The mass of the bullet is one-fourth the mass of the rod. What is thefinal angular speed of the rod?
The value of final angular speed of the uniform rod which rests on the frictionless horizontal surface is,
What is angular speed of a body?
The angular speed of a body is the rate by which the body changed its angle with respect to the time. It can be given as,
A uniform rod of length L rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end.
The rod is initially at rest. A bullet traveling parallel to the horizontal surface and perpendicular to the rod with speed v strikes the rod at its center and becomes embedded in it.
The mass of the bullet is one-fourth the mass of the rod. The diagram for the above condition is attached below.
In the attached image the angular momentum about the point A is constant just before and after the collision. Thus,
Put the value of inertia as,
Solving it further we get,
Hence, the value of final angular speed of the uniform rodwhich rests on the frictionless horizontal surface is,
The concentration of Biochemical Oxygen Demand (BOD) in a river just downstream of a wastewater treatment plant’s effluent pipe is 75 mg/L. If the BOD is destroyed through a first-order reaction with a rate constant equal to 0.05/day, what is the BOD concentration 50 km downstream? The velocity of the river is 15 km/day.
The BOD concentration 50 km downstream when the velocity of the river is 15 km/day is 63.5 mg/L
Let the initial concentration of the BOD = C₀
Concentration of BOD at any time or point = C
dC/dt = - KC
∫ dC/C = -k ∫ dt
Integrating the left hand side from C₀ to C and the right hand side from 0 to t
In (C/C₀) = -kt + b (b = constant of integration)
At t = 0, C = C₀
In 1 = 0 + b
b = 0
In (C/C₀) = - kt
(C/C₀) = e⁻ᵏᵗ
C = C₀ e⁻ᵏᵗ
C₀ = 75 mg/L
k = 0.05 /day
C = 75 e⁻⁰•⁰⁵ᵗ
So, we need the BOD concentration 50 km downstream when the velocity of the river is 15 km/day
We calculate how many days it takes the river to reach 50 km downstream
Velocity = (displacement/time)
15 = 50/t
t = 50/15 = 3.3333 days
So, we need the C that corresponds to t = 3.3333 days
C = 75 e⁻⁰•⁰⁵ᵗ
0.05 t = 0.05 × 3.333 = 0.167
C = 75 e⁻⁰•¹⁶⁷
C = 63.5 mg/L
The BOD concentration 50 km downstream from the wastewater treatment plant is approximately 15.865 mg/L.
To calculate the BOD concentration 50 km downstream, we need to consider the rate of dilution due to the flow of the river and the first-order reaction that destroys BOD. The concentration of BOD downstream can be calculated using the equation C2 = C1 * exp(-k * d/v), where C1 is the initial concentration, k is the rate constant, d is the distance, and v is the velocity of the river.
Plugging in the given values, we have C2 = 75 * exp(-0.05 * 50/15), which gives us a BOD concentration of approximately 15.865 mg/L 50 km downstream from the wastewater treatment plant.
An aluminum rod is 10.0 cm long and a steel rod is 80.0 cm long when both rods are at a temperature of 15°C. Both rods have the same diameter. The rods are now joined end-to-end to form a rod 90.0 cm long. If the temperature is now raised from 15°C to 90°C, what is the increase in the length of the joined rod? The coefficient of linear expansion of aluminum is 2.4 × 10-5 K-1 and that of steel is 1.2 × 10-5 K-1.
The computation in the increase in the length of the joined rod is shown below:
As we know that
Increase in length = increase in the length of aluminum rod + increase in The length of steel rod
= 0.9 cm
We simply added the length of aluminium rod and length of steel rod so that the length of the joined rod could come and the same is to be considered
The increase in length of the joined rod when the temperature is raised from 15°C to 90°C is 0.090 cm.
To determine the increase in length of the joined rod when the temperature is raised from 15°C to 90°C, we need to use the formula for linear expansion: AL = aLAT, where AL is the change in length, AT is the change in temperature, and a is the coefficient of linear expansion. First, we need to calculate the change in temperature for each rod: ΔT = 90°C - 15°C = 75°C. For the aluminum rod, using a linear expansion coefficient of 2.4 × 10-5 K-1 and a length of 10.0 cm, we can calculate the change in length using the formula: ALaluminum = (2.4 × 10-5 K-1)(10.0 cm)(75°C) = 0.018 cm. Similarly, for the steel rod, using a linear expansion coefficient of 1.2 × 10-5 K-1 and a length of 80.0 cm, we can calculate the change in length: ALsteel = (1.2 × 10-5 K-1)(80.0 cm)(75°C) = 0.072 cm. Since the rods are joined end-to-end, the total change in length of the joined rod is the sum of the individual changes: ΔL = ALaluminum + ALsteel = 0.018 cm + 0.072 cm = 0.090 cm. Therefore, the increase in the length of the joined rod is 0.090 cm.
Exposure to what type of radiant energy is sensed by human skin as warmth? x-rays ultraviolet infrared gamma rays
infrared radiant energy is sensed by human skin as warmth. Hence option C is correct.
What is radiation ?
Radiation in physics is the emission or transmission of energy as waves, particles, or both, via space or a material medium. This comprises:
electromagnetic radiation, which includes gamma radiation, x-rays, microwaves, infrared, visible light, and ultraviolet radiation
Particle radiation includes beta radiation, proton radiation, neutron radiation, and other particles with non-zero rest energies.
ultrasonography, sound, and seismic waves (reliant on a physical transmission medium) are examples of acoustic radiation.
gravity radiation, which manifests as gravitational waves or ripples in spacetime's curvature
Depending on the energy of the emitted particles, radiation is frequently divided into ionising and non-ionizing categories. More than 10 eV is carried by ionising radiation, which is sufficient to ionise atoms, molecules.
You observe three carts moving to the right. Cart A moves to the right at nearly constant speed. Cart B moves to the right, gradually speeding up. Cart C moves to the right, gradually slowing down. Which cart or carts, if any, experience a net force to the right
Cart A is moving to the right with constant speed i.e. net acceleration is zero
because acceleration is change in velocity in given time
Cart B is moving towards right with gradually speed up so there is net acceleration which helps to increase the velocity s
This indicates the net force acting on the cart towards right
For cart C there is gradual slow down of cart which indicates cart is decelerating and a net force is acting towards which opposes its motion.
A sample that includes important subgroups that the researchers want to be able to generalize their results to is called
Answer: representative sample
A representative sample can be defined as a subset of the population. This sample includes subgroups that reflect the features of the larger group and are chosen for a specific purpose of study. For example, a classroom with 60 students having 30 females and 30 males, can generate a representative sample based on the gender differences.