A force \(F =\left(5+3 y^2\right)\) acts on a particle in the \(y\)-direction, where \(F\) is in newton and \(y\) is in meter The work done by the force during a displacement from \(y=2 m\) to \(y=5 m\) is___ \(J\).
A force \(F =\left(5+3 y^2\right)\) acts on a particle in the \(y\)-direction, where \(F\) is in newton and \(y\) is in meter The work done by the force during a displacement from \(y=2 m\) to \(y=5 m\) is___ \(J\).
Show Hint
Work done by a variable force is calculated by integrating the force with respect to displacement. Ensure the force and displacement are in the same direction.
Correct Answer: 132
Solution and Explanation
Step 1: Recall the Formula for Work Done
The work done (\(W\)) by a variable force \(F(y)\) over a displacement from \(y_1\) to \(y_2\) is given by the integral:
\[ W = \int_{y_1}^{y_2} F(y) \, dy \]
Step 2: Substitute the Given Force and Limits
In this case, \(F(y) = 5 + 3y^2\), \(y_1 = 2 \, \text{m}\), and \(y_2 = 5 \, \text{m}\). So,
\[ W = \int_{2}^{5} (5 + 3y^2) \, dy \]
Step 3: Evaluate the Integral
\[ W = \left[ 5y + \frac{3y^3}{3} \right]_2^5 = \left[ 5y + y^3 \right]_2^5 \] \[ W = (5(5) + 5^3) - (5(2) + 2^3) \] \[ W = (25 + 125) - (10 + 8) \] \[ W = 150 - 18 = 132 \, \text{J} \]
Conclusion: The work done by the force is \(132 \, \text{J}\).
Top Questions on Random Variables
- If the vector equation of the line \[ \frac{x - 2}{2} = \frac{2y - 5}{-3} = z + 1, \] is given by: \[ \vec{r} = \left(2\hat{i} + \frac{5}{2}\hat{j} - \hat{k}\right) + \lambda\left(2\hat{i} - \frac{3}{2}\hat{j} + p\hat{k}\right), \] then \( p \) is equal to:
- MHT CET - 2024
- Mathematics
- Random Variables
- The integral \( \int e^x \frac{2 + \sin 2x}{1 + \cos 2x} \, dx \) is equal to
- MHT CET - 2024
- Mathematics
- Random Variables
A random sample of size $5$ is taken from the distribution with density \[ f(x;\theta)= \begin{cases} \dfrac{3x^2}{\theta^3}, & 0[6pt] 0, & \text{elsewhere}, \end{cases} \] where $\theta$ is unknown. If the observations are $3,6,4,7,5$, then the maximum likelihood estimate of the $1/8$ quantile of the distribution (rounded off to one decimal place) is __________.
- GATE ST - 2022
- Estimation
- Random Variables
- Let $0,1,1,2,0$ be five observations of a random variable $X$ which follows a Poisson distribution with parameter $\theta>0$. Let the minimum variance unbiased estimate of $P(X\le 1)$, based on this data, be $\alpha$. Then $5^{4}\alpha$ (in integer) is equal to $\;__________$
- GATE ST - 2022
- Estimation
- Random Variables
- Consider a gamma distribution with pdf \[ f(x;\beta)= \begin{cases} \dfrac{1}{24\beta^{5}}x^{4}e^{-x/\beta}, & x>0,\\ 0, & \text{elsewhere}, \end{cases} \] with $\beta>0$. For $\beta=2$, find the Cramér–Rao lower bound (rounded to one decimal place) for the variance of any unbiased estimator of $\beta^{2}$ based on a random sample of size 8.
- GATE ST - 2022
- Estimation
- Random Variables
Questions Asked in JEE Main exam
- If the distances of the point \( (1,2,a) \) from the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] along the lines \[ L_1:\ \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} \quad \text{and} \quad L_2:\ \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} \] are equal, then \( a+b+c \) is equal to:
- JEE Main - 2026
- Three Dimensional Geometry
Two circular discs of radius \(10\) cm each are joined at their centres by a rod, as shown in the figure. The length of the rod is \(30\) cm and its mass is \(600\) g. The mass of each disc is also \(600\) g. If the applied torque between the two discs is \(43\times10^{-7}\) dyne·cm, then the angular acceleration of the system about the given axis \(AB\) is ________ rad s\(^{-2}\).
- JEE Main - 2026
- Rotational motion
- In an experiment, a set of readings are obtained as follows: \[ 1.24~\text{mm},\ 1.25~\text{mm},\ 1.23~\text{mm},\ 1.21~\text{mm}. \] The expected least count of the instrument used in recording these readings is _______ mm.
- JEE Main - 2026
- General Physics
Method used for separation of mixture of products (B and C) obtained in the following reaction is:
- JEE Main - 2026
- p -Block Elements
- The number of numbers greater than $5000$, less than $9000$ and divisible by $3$, that can be formed using the digits $0,1,2,5,9$, if repetition of digits is allowed, is
- JEE Main - 2026
- Permutations
Concepts Used:
Work, Energy and Power
Work:
- Work is correlated to force and the displacement over which it acts. When an object is replaced parallel to the force's line of action, it is thought to be doing work. It is a force-driven action that includes movement in the force's direction.
- The work done by the force is described to be the product of the elements of the force in the direction of the displacement and the magnitude of this displacement.
Energy:
- A body's energy is its potential to do tasks. Anything that has the capability to work is said to have energy. The unit of energy is the same as the unit of work, i.e., the Joule.
- There are two types of mechanical energy such as; Kinetic and potential energy.
Read More: Work and Energy
Power:
- Power is the rate at which energy is transferred, conveyed, or converted or the rate of doing work. Technologically, it is the amount of work done per unit of time. The SI unit of power is Watt (W) which is joules per second (J/s). Sometimes the power of motor vehicles and other machines is demonstrated in terms of Horsepower (hp), which is roughly equal to 745.7 watts.
- Power is a scalar quantity, which gives us a quantity or amount of energy consumed per unit of time but with no manifestation of direction.