Question
Question: Mass \({m_1}\) hits and sticks with \({m_2}\) while sliding horizontally with velocity \(v\) along t...
Mass m1 hits and sticks with m2 while sliding horizontally with velocity v along the common line of centres of three equal masses (m1=m2=m3=m). Initially masses m2 and m3 are stationary and the spring is unstretched. Minimum kinetic energy of m2 is 36ymv2. Find y.
Solution
Hint
In this problem, there are two types of energy. First the kinetic energy of the first mass transferred to the second mass. So, the kinetic energy of the two masses gives the potential energy to the spring, so by equating the kinetic energy of the two masses with the potential energy of the spring, the solution can be determined.
The kinetic energy is given by,
⇒KE=21mv2
Where, KE is the kinetic energy, m is the mass of the block and v is the velocity of the block.
The potential energy of the spring is,
⇒PE=21kx2
Where, PE is the potential energy of the spring, k is the spring constant and x is the compression or expansion of the spring.
Complete step by step answer
Given that, The mass m1 hits the mass m2, so that the velocity of the mass m1 is reduced by half.
Three masses are equal (m1=m2=m3=m).
When the mass m1 hits the mass m2 and the mass m2 compresses the spring, then the kinetic energy of the two masses is equal to the potential energy of the spring. Then,
⇒21(2m)(2v)2=21kx2
Both the masses are equal so 2m and the velocity is reduced by half when it hits the second mass so 2v.
From the above equation,
⇒21(2m)4v2×k2=x2
By cancelling the terms, then
⇒2kmv2=x2
The above equation is written as,
⇒x=2kmv2
Since, the compression of the spring is 32 times of maximum of m3 from the centre of mass, then the kinetic energy of the m3 gets from the potential energy of the spring, then
Kinetic energy of m3 is ⇒21k(32x)2
Substituting the value of x in the above equation, then
⇒21k(32×(2kmv2))2
By squaring the terms, then
⇒21×k×94×2kmv2
By cancelling the terms, then
⇒9mv2
Here multiplying and dividing by 4, then
⇒9mv2×44
On multiplying the above equation, then
⇒364mv2
By comparing the term given in the question 36ymv2, then the value of y is, y=4.
Note
The energy can neither be created nor destroyed. The kinetic energy of the first mass gives the kinetic energy to the second mass, and the second mass gives the potential energy to the spring. That potential energy of the spring gives again kinetic energy to the third mass. Thus, the energy gets transferred but not created.