Two masses m1 and m2 move in a uniform gravitational field g and interact via a potential energy U(r).
(a) Show that the Lagrangian can be decomposed as
L = L_cm + L_rel
(b) Write down the Lagrange equations for the CM coordinates X, Y, Z and describe the motion of the CM. Then write the Lagrange equations for the relative coordinates and show that the motion of r is the same as that of a particle of reduced mass μ, position r, and potential energy U(r).
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SOL:
r1, r2: positions
BCD is a way to represent decimal numbers in binary form, where each decimal digit is represented by a fixed number of bits, usually four.
Benefits of BCD include:
Straightforward to convert between BCD and denary (base-10).
Less complex to encode and decode for programmers.
Easier for digital equipment to use BCD to display information.
Can represent monetary values exactly.
Applications of BCD:
Electronic displays
A: Profit is the reward earned by an entrepreneur for organizing and managing the
factors of production. Economists have given various theories to explain the origin and
nature of profit. Each theory highlights a di erent aspect of entrepreneurship. The six
major theories of profit are explained below:
1. Rent Theory of Profit – F.A. Walker
This theory compares profit to rent.
Just like landowners get rent for using more fertile land, entrepreneurs get profit
due to their superior ability.
SECTION 6.1 - Data, Information, and Business Intelligence
Four Primary Traits of Data Value:
Accuracy – Correct and reliable.
Completeness – No missing values.
Consistency – Same data, same format across systems.
Timeliness – Up-to-date and available when needed.
Types of Data:
Transactional Data: Supports daily tasks (e.g., sales receipts).
Analytical Data: Supports managerial decisions (e.g., trends).
Real-Time
Chapter 17: The Special Senses
Olfaction (Smell)
Gustation (Taste)
Vision...
I. CLASSIFICATION EVALUATION METRICS
A. Confusion Matrix (for two classes: Positive (+), Negative (-))
Predicted + Predicted -
Actual + TP (f++) FN (f+-)
Actual - FP (f-+) TN (f--)
True Positive (TP): Actual +, Predicted +. Correctly identified positives.
False Negative (FN): Actual +, Predicted -. Missed positives (Type II error).
False Positive (FP): Actual -, Predicted +. Falsely identified positives (Type I error).
True Negative (TN): Actual -, Predicted -. Correctly identified negatives.
Total Actual...
1) import matplotlib.pyplot as plt||import pandas as pd||import seaborn as sns||from sklearn.datasets import fetch_california_housing||# Load the California Housing dataset||california = fetch_california_housing(as_frame=True)||df = california.frame||# Numerical features||numerical_features = df.select_dtypes(include=['number']).columns||# Histograms||plt.figure(figsize=(15, 10))||for i, feature in enumerate(numerical_features):||plt.subplot(3, 3, i + 1)||sns.histplot(df[feature], kde=True)||plt.
1. Introduction: Lord Ripon, the Viceroy of India from 1880 to 1884, is known as the "Father of Local Self-Government" in India.
He believed in giving Indians a role in administration and improving governance through local bodies.
2. Main Objectives:To promote democratic participation at the local level.
To make administration more efficient and responsive to local needs.To train Indians in the art of self-governance.
3. Key Features of the 1882 Resolution: Establishment of Local Boards: District...
1. useState
Purpose: Adds state to functional components.
Import: import { useState } from 'react';
Syntax: const [state, setState] = useState(initialValue);
Update: setState(newValue); or setState(prevState => ...); (for updates based on previous state).
Example:
import React, { useState } from 'react';
function Counter() {
const [count, setCount] = useState(0);
return (
<div>
<p>Count: {count}</p>
<button onClick={() => setCount(count + 1)}>Increment</button>
</div>...