A bullet of mass ‘m’ traveling horizontally at a speed ‘v’ embeds itself in the centre of a block of wood of mass ‘M’, which is suspended by a light vertical string of length ‘l’. What would be the maximum inclination of the string with the vertical?
After the bullet gets embedded in the block, the block acquires velocity V given by
mv = (m+M) V
V = [m/(m+M)] v
Then the pendulum swings to a height H while total energy is conserved:
gH = V^2/2 = [m/(m+M)]^2 v^2/2
If the maximum inclination angle of the swing is A and the string length is l,
l (1 - cos A) = H
Solve for A
To find the maximum inclination of the string, we need to consider the conservation of energy.
First, let's assume that the block of wood remains at rest after the bullet embeds itself in it. In this case, the total initial kinetic energy of the system (bullet + block) is given by:
KE_initial = (1/2) * m * v^2
Given that the bullet embeds itself in the center of the block, the system's kinetic energy is transferred into potential energy due to the change in height of the center of mass of the block.
The change in height, Δh, can be determined by observing the change in length of the string, which is equal to l * sin(θ), where θ is the angle of inclination of the string (with the vertical).
The change in potential energy, ΔPE, can be calculated as the product of the mass of the block (M) and the acceleration due to gravity (g) and the change in height (Δh):
ΔPE = M * g * Δh = M * g * l * sin(θ)
Since the system starts with its initial kinetic energy being converted to the potential energy at the maximum inclination angle, the change in potential energy ΔPE should be equal to the kinetic energy initially possessed by the system.
Therefore, we can equate these two expressions:
ΔPE = KE_initial
M * g * l * sin(θ) = (1/2) * m * v^2
Now, we can solve this equation to find the maximum inclination angle θ:
sin(θ) = (1/2) * (m/M) * (v^2 / (g * l))
Taking the inverse sine (arcsine) of both sides, we get:
θ = arcsin((1/2) * (m/M) * (v^2 / (g * l)))
Hence, the maximum inclination of the string with the vertical, θ, can be found using the above equation.