Two blocks of masses $m$ and $M$, ($M > m$), are placed on a frictionless table as shown in figure. A massless spring with spring constant $k$ is attached with the lower block. If the system is slightly displaced and released, then A. The time period of small oscillation of the two blocks is $T = 2\pi \sqrt{\frac{(m+M)}{k}}$ B. The acceleration of the blocks is $a = \frac{kx}{M+m}$ (x = displacement of the blocks from the mean position) C. The magnitude of the frictional force on the upper block is $\frac{\mu m|x|}{M+m}$ D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu (M+m)g}{k}$ E. Maximum frictional force can be $\mu(M + m) g$. Choose the correct answer from the options given below:
A, B, D Only
Let's analyze each statement to determine the correct options:
- Time Period of Small Oscillations:
For a system with two masses connected by a spring, the time period T is given by:
T = 2\pi \sqrt{\frac{(m+M)}{k}}
This formula is derived from the definition of simple harmonic motion (SHM) where k is the spring constant and m+M is the total mass. This statement is correct.
- Acceleration of the Blocks:
The acceleration a in SHM when displaced by x is:
a = \frac{kx}{m + M}
This is correct for a spring-mass system experiencing SHM. Therefore, this statement is correct.
- Frictional Force on Upper Block:
The frictional force f on the upper block cannot be calculated simply as \frac{\mu m|x|}{M+m}. The friction on the upper block depends on whether it is moving or not, and the maximum static friction is \mu mg. Therefore, statement C is incorrect.
- Maximum Amplitude without Slipping:
Maximum amplitude A for which the block does not slip is given by equating maximum frictional force to the force required for maximum amplitude oscillation:
A = \frac{\mu(M+m)g}{k}
This statement is correct.
- Maximum Frictional Force:
The maximum frictional force is correctly defined as \mu mg. This alone does not incorporate M. Therefore, statement E is incorrect.
Therefore, the correct options are A, B, and D only.
Similar Questions
A wire of length $25 \ m$ and cross-sectional area $5 \ mm^2$ having resistivity of $2 \times 10^{-6} \ \Omega \ m$ is bent into a complete circle. The resistance between diametrically opposite points will be
(DROPPED)
A parallel plate capacitor is filled equally(half) with two dielectrics of dielectric constants $\varepsilon_1$ and $\varepsilon_2$, as shown in figures. The distance between the plates is $d$ and area of each plate is $A$. If capacitance in first configuration and second configuration are $C_1$ and $C_2$ respectively, then $\frac{C_1}{C_2}$ is :
First Configuration
The electrostatic potential on the surface of uniformly charged spherical shell of radius $R = 10 \ cm$ is $120 \ V$. The potential at the centre of shell, at a distance $r = 5 \ cm$ from centre, and at a distance $r = 15 \ cm$ from the centre of the shell respectively, are:
The radiation pressure exerted by a $450 \ W$ light source on a perfectly reflecting surface placed at $2m$ away from it, is
A loop ABCDA, carrying current I = 12 A, is placed in a plane, consists of two semi-circular segments of radius $R_1 = 6\pi$ m and $R_2 = 4\pi$ m. The magnitude of the resultant magnetic field at center O is $k\times10^{-7}$ T. The value of k is __________
(Given $\mu_0 = 4\pi \times 10^{-7}$ Tm $A^{-1}$)
A small bob of mass 100 mg and charge $+10 \text{ }\mu C$ is connected to an insulating string of length 1 m. It is brought near to an infinitely long non-conducting sheet of charge density '$\sigma$' as shown in figure. If string subtends an angle of $45^\circ$ with the sheet at equilibrium the charge density of sheet will be.
(Given, $\epsilon_0 = 8.85\times10^{-12} \frac{F}{m}$ and acceleration due to gravity, $g=10 \frac{m}{s^2}$)
- The battery of a mobile phone is rated as 4.2 V, 5800 mAh. How much energy is stored in it when fully charged ?
The relationship between the magnetic susceptibility ($\chi$) and the magnetic permeability ($\mu$) is given by :
($\mu_0$ is the permeability of free space and $\mu_r$ is relative permeability)Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density $+ \sigma$ and $- 2\sigma$. The force experienced by a point charge $+ q$ placed at the mid point between two plates will be :
- Let $B_1$ be the magnitude of magnetic field at center of a circular coil of radius $R$ carrying current $I$. Let $B_2$ be the magnitude of magnetic field at an axial distance '$x$' from the center. For $x:R=3:4$, $\frac{B_2}{B_1}$ is:
Important Questions from Electricity and Magnetism
A wire of length $25 \ m$ and cross-sectional area $5 \ mm^2$ having resistivity of $2 \times 10^{-6} \ \Omega \ m$ is bent into a complete circle. The resistance between diametrically opposite points will be
(DROPPED)
A parallel plate capacitor is filled equally(half) with two dielectrics of dielectric constants $\varepsilon_1$ and $\varepsilon_2$, as shown in figures. The distance between the plates is $d$ and area of each plate is $A$. If capacitance in first configuration and second configuration are $C_1$ and $C_2$ respectively, then $\frac{C_1}{C_2}$ is :
First Configuration
The electrostatic potential on the surface of uniformly charged spherical shell of radius $R = 10 \ cm$ is $120 \ V$. The potential at the centre of shell, at a distance $r = 5 \ cm$ from centre, and at a distance $r = 15 \ cm$ from the centre of the shell respectively, are:
The radiation pressure exerted by a $450 \ W$ light source on a perfectly reflecting surface placed at $2m$ away from it, is
A loop ABCDA, carrying current I = 12 A, is placed in a plane, consists of two semi-circular segments of radius $R_1 = 6\pi$ m and $R_2 = 4\pi$ m. The magnitude of the resultant magnetic field at center O is $k\times10^{-7}$ T. The value of k is __________
(Given $\mu_0 = 4\pi \times 10^{-7}$ Tm $A^{-1}$)
