A mass m_1 connected to a horizontal spring p | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) At the mean position,the potential energy is zero and the kinetic energy is maximum. The velocity $v$ of mass $m_1$ at the mean position is $v = A

Vedclass · Accepted Answer

$\left[\frac{m_1}{m_1 + m_2}\right]^{\frac{1}{2}}$

Vedclass · Answer

(A) At the mean position,the potential energy is zero and the kinetic energy is maximum. The velocity $v$ of mass $m_1$ at the mean position is $v = A\omega = A\sqrt{\frac{k}{m_1}}$.When mass $m_2$ is placed on $m_1$,the momentum is conserved because the spring force is zero at the mean position. Let $v'$ be the new velocity:$m_1 v = (m_1 + m_2) v'$$v' = \frac{m_1 v}{m_1 + m_2} = \frac{m_1 A \sqrt{k/m_1}}{m_1 + m_2} = A \sqrt{\frac{k m_1}{(m_1 + m_2)^2}}$.The new energy of the system is $E' = \frac{1}{2} k A_1^2 = \frac{1}{2} (m_1 + m_2) (v')^2$.Substituting $v'$:$\frac{1}{2} k A_1^2 = \frac{1}{2} (m_1 + m_2) \left( A^2 \frac{k m_1}{(m_1 + m_2)^2} \right)$.$A_1^2 = A^2 \frac{m_1}{m_1 + m_2}$.Therefore,$\frac{A_1}{A} = \sqrt{\frac{m_1}{m_1 + m_2}}$.

$A$ mass $m_1$ connected to a horizontal spring performs $S.H.M.$ with amplitude $A$. While mass $m_1$ is passing through the mean position,another mass $m_2$ is placed on it so that both the masses move together with amplitude $A_1$. The ratio of $\frac{A_1}{A}$ is:

Similar Questions

$A$ body executes $SHM$ under the action of force $F_1$ with time period $T_1$. If the force is changed to $F_2$,it executes $SHM$ with time period $T_2$. If both the forces $F_1$ and $F_2$ act simultaneously in the same direction on the body,its time period is:

Two identical springs of constant $K$ are connected in series and parallel as shown in the figure. $A$ mass $m$ is suspended from them. The ratio of their frequencies of vertical oscillations will be

$A$ cylindrical block of density $\rho$ is partially immersed in a liquid of density $3\rho$. The plane surface of the block remains parallel to the surface of the liquid. The height of the block is $60\, cm$. The block performs $SHM$ when displaced from its mean position. [Use $g = 9.8\, m/s^2$]

Two particles $A$ and $B$ of equal masses are suspended from two massless springs of spring constants $k_1$ and $k_2$,respectively. If the maximum velocities during oscillations are equal,the ratio of amplitudes of $A$ and $B$ is

On a smooth inclined plane,a mass $M$ is attached between two massless springs of force constant $k$ each,as shown in the figure. The other ends of the springs are fixed to firm supports. The period of oscillation of the mass $M$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module