Question:
A particle moves along x-axis under force \(F(x) = -kx + ax^2\). Find nature of potential energy graph.
A particle moves along x-axis under force \(F(x) = -kx + ax^2\). Find nature of potential energy graph.
Show Hint
Force is negative slope of potential → integrate carefully to get graph shape.
Updated On: Apr 23, 2026
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
\[
F = -\frac{dU}{dx}
\]
Step 1: Find potential energy
\[ \frac{dU}{dx} = kx - ax^2 \] \[ U = \frac{kx^2}{2} - \frac{ax^3}{3} \]
Step 2: Analyze shape
• Quadratic term $\Rightarrow$ upward curve initially
• Cubic term $\Rightarrow$ bends downward later So curve rises, then falls $\Rightarrow$ matches option (D) Conclusion: \[ {(D)} \]
Step 1: Find potential energy
\[ \frac{dU}{dx} = kx - ax^2 \] \[ U = \frac{kx^2}{2} - \frac{ax^3}{3} \]
Step 2: Analyze shape
• Quadratic term $\Rightarrow$ upward curve initially
• Cubic term $\Rightarrow$ bends downward later So curve rises, then falls $\Rightarrow$ matches option (D) Conclusion: \[ {(D)} \]
Was this answer helpful?
0
0
Top MET Physics Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- A rope of mass $0.1\, kg$ is connected at the same height of two opposite walls. It is allowed to hang under its own weight. At [he contact point between the rope and the wall, the rope makes an angle $ \theta =10^{\circ} $ with respect to horizontal. The tension in the rope at its midpoint between the walls is
- At great distances from an electric dipole, the electric field strength due to the dipole varies with the distance as
View More Questions
Top MET work, energy and power Questions
- The escape velocity from the surface of the earth is \( v_{e} \). The escape velocity from the surface of a planet whose mass and radius are 3 times those of the earth will be:
- The potential energy of a simple harmonic oscillator at its mean position is:
- The escape velocity of a body from the surface of the earth is 11.2 km/s. If a body is projected with a velocity twice the escape velocity, then the velocity of the body far away from the earth will be:
- The power of a water pump is 2 kW. If \( g = 10 \text{ m/s}^{2} \), the amount of water it can raise in one minute to a height of 10 m is:
- A body of mass 8 kg is moved by a force \(F = 3x\) N, where x is the distance covered. Initial position is x = 2 m and the final position is x = 10 m. The initial speed is zero. The final speed is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions