A block A of mass m 1 and block B of mass m 2 are connected together by a spring and placed on a bearing surface as shown- The block A performs free vertical harmonic oscillations with amplitude A max and frequency - Assuming that block B doesn-t leave the surface and neglecting the mass of the spring- find the maximum and minimum value of force R max- R min- that the system exerts on the bearing surface-R max- R min- m 1- m 2 g - m 1-2 A max

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Simple Harmonic Oscillation

A block A of ...

Question

A block $A$ of mass $m_{1}$ and block $B$ of mass $m_{2}$ are connected together by a spring and placed on a bearing surface as shown. The block $A$ performs free vertical harmonic oscillations with amplitude $A_{m a x}$ and frequency $ω$ . Assuming that block $B$ doesn't leave the surface and neglecting the mass of the spring, find the maximum and minimum value of force ( $R_{m a x} & R_{m i n}$ ) that the system exerts on the bearing surface.

R_{max} = (m_{1} + m_{2}) g + m_{1} ω^{2} A_{m a x}; R_{min} = (m_{1} + m_{2}) g - m_{1} ω^{2} A_{m a x}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

R_{max} = (m_{1} + m_{2}) g + m_{2} ω^{2} A_{m a x}; R_{min} = (m_{1} + m_{2}) g - m_{2} ω^{2} A_{m a x}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

R_{max} = R_{min} = (m_{1} + m_{2}) g + m_{1} ω^{2} A_{m a x}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

R_{max} = R_{min} = (m_{1} + m_{2}) g - m_{1} ω^{2} A_{m a x}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $R_{max} = (m_{1} + m_{2}) g + m_{1} ω^{2} A_{m a x}; R_{min} = (m_{1} + m_{2}) g - m_{1} ω^{2} A_{m a x}$
As $A$ oscillates up and down, the normal reaction between $B$ and the surface is maximum when $A$ is at the lowest position and it is minimum when $A$ is at the topmost position.

Case I: Lowest position of $A$
Let $T_{1}$ be the force in the compressed spring.
As $B$ is at rest, we can say that,
$T_{1} + m_{2} g = R_{max} . . . . . . . . (1)$
As acceleration of $A$ is $a_{m a x} = ω^{2} A_{m a x}$ towards mean position $(↑)$ (where $A_{m a x}$ is amplitude),
$T_{1} - m_{1} g = m_{1} ω^{2} A_{m a x} . . . . . . . (2)$
From $(1)$ and $(2)$ ,
$R_{m a x} = (m_{1} + m_{2}) g + m_{1} ω^{2} A_{m a x} . . . . . . (3)$

Case II: Topmost position of $A$

Let $T_{2}$ be the force in the elongated spring.
As $B$ is at rest,
$T_{2} + R_{min} = m_{2} g$
Acceleration of $A$ is $a_{m a x} = ω^{2} A_{m a x}$ towards mean position $(↓)$ , where $A_{m a x}$ is amplitude.
$T_{2} + m_{1} g = m_{1} ω^{2} A_{m a x}$
Combining, we get $R_{min} = (m_{1} + m_{2}) g - m_{1} ω^{2} A_{m a x} . . . . . . (4)$
From $(3)$ and $(4)$ , we can say that option (a) is the correct answer.

Suggest Corrections

Similar questions

Q. The blocks of masses

m_{1}

and

m_{2}

are connected by an ideal spring of force constant k. The blocks are placed on a smooth horizontal surface. A horizontal force F acts on the block

m_{1}

. Initially, spring is relaxed, both the blocks are at rest.

What is the maximum elongation of spring?

Q. Two blocks of masses

M_{1}

and

M_{2}

are connected with a string passing over a pulley as shown in the figure. The block

M_{1}

lies on a horizontal surface. The coefficient of friction between the block

M_{1}

and the horizontal surface is

μ

. The system accelerates. What additional mass

m

should be placed on the block

M_{1}

so that the system does not accelerate?

Q. Two blocks of masses

m_{1}

and

m_{2}

are connected by a massless spring and placed on smooth surface. The spring initially stretched and released. Then :

A 1 kg block is executing simple harmonic motion of amplitude 0.1 m on a smooth horizontal surface under the restoring force of a spring of spring constant $100 N m^{- 1.}$ A block of mass 3 kg is gently placed on it at the instant it passes through the mean position. Assuming that the two blocks move together, find the frequency and the amplitude of the motion.