A constant power is supplied to a rotating disc. The relationship between the angular velocity $\omega$ of the disc and number of rotations (n) made by the disc is governed by
- $\omega\propto\,n^{\frac{1}{3}}$
- $\omega\propto\,n^{\frac{2}{3}}$
- $\omega\propto\,n^{\frac{3}{2}}$
- $\omega\propto\,n^2$
The Correct Option is A
Solution and Explanation
Rotational energy \(= \frac 12 Iω^2\)
Rotational power \(= \frac {d}{dt} (\frac 12 Iω^2)\)
Given that Rotational power is constant,
\(⇒ \frac {d}{dt} (\frac 12 Iω^2) = K\) (Constant)
\(⇒ ∫d (\frac 12 Iω^2) = ∫K dt\)
\(⇒ \frac 12 Iω^2 = Kt\)
So, \(ω^2 ∝ t \) …….. (1)
But we know that,
\(ω = \frac {2\pi rn}{t}\) ……. (2)
From eq (1) and (2),
\(ω^2 ∝ \frac nω\)
\(⇒ ω^3 ∝ n\)
\(⇒ ω ∝ n^{\frac 13}\)
So, the correct option is (A): \(ω ∝ n^{\frac 13}\)
Top Questions on Vectors
- If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]
- A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3\hat{i} + 15\hat{j} + 6\hat{k}$ and the other is along the vector $2\hat{i} + 10\hat{j} + \hat{k}$, then the value of $\lambda$ is :
- A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \( \hat{i} + 15 \hat{j} + 6 \hat{k} \) and the other is along the vector \( 2 \hat{i} + 10 \hat{j} + 6 \hat{k} \), then the value of \( \lambda \) is :
Show that the line passing through the points A $(0, -1, -1)$ and B $(4, 5, 1)$ intersects the line joining points C $(3, 9, 4)$ and D $(-4, 4, 4)$.
- Let $|\vec{a}| = 5 \text{ and } -2 \leq \lambda \leq 1$. Then, the range of $|\lambda \vec{a}|$ is:
Questions Asked in CBSE CLASS XII exam
- Determine those values of $x$ for which $f(x) = \frac{2}{x} - 5$, $x \ne 0$ is increasing or decreasing.
- CBSE CLASS XII - 2025
- Functions
- Agro-startups and rural enterprises are becoming increasingly vital for rural development in India. They address the critical challenges faced by rural sector in India. These ventures bring in technological innovations such as precision farming, IT-based monitoring system and data-driven decision making tools that enhance productivity and resource efficiency. By streamlining supply chains and creating direct market linkages, they ensure better pricing for farmers and reduce dependency on traditional intermediaries. Such enterprises contribute actively to employment generation in the rural areas. The government has demonstrated a continuous policy and financial support for agri-startups and entrepreneurs through various programs like RKVY-RAFTAAR, the Agri- infrastructure fund and Agri-SURE.
On the basis of given text and common understanding, answer the following questions:- CBSE CLASS XII - 2025
- Rural Development
Read the following text carefully:
Union Food and Consumer Affairs Minister said that the Central Government has taken many proactive steps in the past few years to control retail prices of food items. He said that the government aims to keep inflation under control without compromising the country’s economic growth. Retail inflation inched up to a three-month high of 5.55% in November 2023 driven by higher food prices. Inflation has been declining since August 2023, when it touched 6.83%. 140 new price monitoring centres had been set up by the Central Government to keep a close watch on wholesale and retail prices of essential commodities. The Government has banned the export of many food items like wheat, broken rice, non-basmati white rice, onions etc. It has also reduced import duties on edible oils and pulses to boost domestic supply and control price rise. On the basis of the given text and common understanding,
answer the following questions:- CBSE CLASS XII - 2025
- Government Budget and the Economy
- Distinguish between Physical Capital and Human Capital.
- CBSE CLASS XII - 2025
- Human Capital Formation
- “In the recent Budget the Government allocated funds for construction of new metro lines in smart cities to improve public transportation and reduce pollution.”
Identify the nature of government expenditure \& its objective.- CBSE CLASS XII - 2025
- Government Budget and the Economy
Concepts Used:
Vectors
The quantities having magnitude as well as direction are known as Vectors or Vector quantities. Vectors are the objects which are found in accumulated form in vector spaces accompanying two types of operations. These operations within the vector space include the addition of two vectors and multiplication of the vector with a scalar quantity. These operations can alter the proportions and order of the vector but the result still remains in the vector space. It is often recognized by symbols such as U ,V, and W
Representation of a Vector :
A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector. It is represented by a or commonly as AB. In this line segment AB, A is the starting point and B is the terminal point of the line.
Types of Vectors:
Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:
- Zero vector
- Unit Vector
- Position Vector
- Co-initial Vector
- Like and Unlike Vectors
- Coplanar Vector
- Collinear Vector
- Equal Vector
- Displacement Vector
- Negative of a Vector