A force \( F = \alpha + \beta x^2 \) acts on an object in the x-direction. The work done by the force is 5 J when the object is displaced by 1 m. If the constant \( \alpha = 1 \, {N} \), then \( \beta \) will be:
Show Hint
- 15 N/m²
- 10 N/m²
- 12 N/m²
- 8 N/m²
The Correct Option is C
Approach Solution - 1
To find the value of \( \beta \), we need to calculate the work done by the force \( F = \alpha + \beta x^2 \) as it displaces an object by 1 meter.
The work done by a force is given by the integral of the force over the displacement:
\(W = \int F \, dx = \int (\alpha + \beta x^2) \, dx\)
Given:
- \( \alpha = 1 \, \text{N} \)
- The work done \( W = 5 \, \text{J} \)
- The displacement is from \( x = 0 \) to \( x = 1 \, \text{m} \)
We now calculate the integral:
\(W = \int_0^1 (1 + \beta x^2) \, dx = \left[ x + \frac{\beta x^3}{3} \right]_0^1\)
Substituting the limits of integration:
\(W = \left( 1 + \frac{\beta}{3} \right) - \left( 0 + 0 \right) = 1 + \frac{\beta}{3}\)
Equating it to the given work done:
\(1 + \frac{\beta}{3} = 5\)
Solve for \( \beta \):
- Subtract 1 from both sides:
\(\frac{\beta}{3} = 4\) - Multiply both sides by 3:
\(\beta = 12\) N/m²
Therefore, the correct value of \( \beta \) is 12 N/m².
This matches the correct answer from the given options.
Approach Solution -2
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Work and Energy Questions
- A particle is placed at the point \( A \) of a frictionless track \( ABC \) as shown in the figure. It is gently pushed toward the right. The speed of the particle when it reaches the point \( B \) is: (Take \( g = 10 \, \text{m/s}^2 \)).
A heavy iron bar of weight 12 kg is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle \(60^\circ\) with the horizontal, the weight experienced by the man is :
- The potential energy function (in joules) of a particle in a region of space is given as\[U = (2x^2 + 3y^3 + 2z).\]Here \( x \), \( y \), and \( z \) are in meters. The magnitude of the \( x \)-component of force (in newtons) acting on the particle at point \( P (1, 2, 3) \) m is:
- An artillery piece of mass \( M_1 \) fires a shell of mass \( M_2 \) horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:
- A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is ______ J.
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.