a uniform rod of mass M is hinged at its upper end. A particle of mass m moving horizontally strikes the rod at its midpoint elastically. If the particle comes to rest after collision find the value of M/m
M/m=1
3/4
To find the value of M/m, we need to analyze the conservation of linear momentum and the conservation of kinetic energy in this collision.
Let's assume that before the collision, the particle of mass m moves with velocity u towards the midpoint of the rod, and after the collision, the particle comes to rest.
Conservation of Linear Momentum:
Since the collision is elastic and there are no external forces acting horizontally, the linear momentum before and after the collision should be conserved.
Before the collision:
Initial momentum of the particle (m) = m * u (to the right)
Initial momentum of the rod (M) = 0 (as it is initially at rest)
After the collision:
Final momentum of the particle (m) = 0 (particle comes to rest)
Final momentum of the rod (M) = M * v (we need to find the final velocity of the rod, v)
Using the law of conservation of linear momentum, we can write the equation:
m * u + 0 = 0 + M * v
Simplifying the equation, we get:
m * u = M * v ---> Equation 1
Conservation of Kinetic Energy:
Since the collision is elastic, the total kinetic energy before and after the collision should be conserved.
Before the collision:
Initial kinetic energy of the particle (m) = (1/2) * m * u^2 (since the particle is moving horizontally)
Initial kinetic energy of the rod (M) = 0 (as it is initially at rest)
After the collision:
Final kinetic energy of the particle (m) = 0 (particle comes to rest)
Final kinetic energy of the rod (M) = (1/2) * M * v^2 (we need to find the final velocity of the rod, v)
Using the law of conservation of kinetic energy, we can write the equation:
(1/2) * m * u^2 + 0 = 0 + (1/2) * M * v^2
Simplifying the equation, we get:
m * u^2 = M * v^2 ---> Equation 2
Dividing Equation 1 by Equation 2, we can solve for M/m:
m * u / (m * u^2) = (M * v) / (M * v^2)
Simplifying and canceling out the common terms, we get:
1 / u = 1 / v
Since the particle comes to rest after collision, v = 0.
Therefore, substituting v = 0 in the equation above, we can solve for u:
1 / u = 1 / 0
Division by zero is undefined. Hence, the value of M/m cannot be determined.
In this case, for the particle to come to rest after the collision, we cannot determine the exact value of M/m.