A bullet of mass 20 g traveling Horizontally at 100m/s,embeds itself in the centre of a block of wood mass 1kg which is suspended by light vertical string 1metre length.calculate the maximum inclination of the string to vertical assuming g=9•8m/s
To calculate the maximum inclination of the string, we need to consider the conservation of momentum. Here are the steps to find the answer:
1. Calculate the momentum of the bullet before it embeds itself in the block of wood.
Momentum = mass × velocity
Given: mass of the bullet (m1) = 20 g = 0.02 kg
velocity of the bullet (v1) = 100 m/s
Momentum of bullet before embedding = m1 × v1
2. Set up an equation for the conservation of momentum.
According to momentum conservation, the momentum before the event should be equal to the momentum after the event.
Momentum before = Momentum after
3. Calculate the momentum of the system (block + bullet) after embedding.
The momentum of the system after embedding will be the momentum of the block and bullet together because they move as one object.
Momentum of the system after embedding = (mass of the block + mass of the bullet) × velocity of the system
4. Equate the two momentum values and solve for the velocity of the system.
Momentum before = Momentum after
m1 × v1 = (m2 + m1) × v2
Where m2 is the mass of the block of wood and v2 is the velocity of the system after embedding.
5. Calculate the velocity of the system (v2).
Rearrange the equation from step 4 to solve for v2.
v2 = (m1 × v1) / (m2 + m1)
6. Calculate the acceleration (a) of the system after embedding.
The acceleration can be calculated using the equation: acceleration = change in velocity / time taken.
As the bullet embeds itself, the system starts from velocity v2 and stops at 0 velocity, so the change in velocity is v2.
Time taken (t) can be calculated as t = 2L / v2, where L is the length of the string (1 m).
7. Calculate the maximum inclination of the string (θ) to vertical.
The maximum inclination occurs when the acceleration is maximum, as given by the equation: a = g × sin(θ)
Rearrange the equation to solve for θ.
θ = arcsin(a / g)
Substitute the calculated values into the equation to get the maximum inclination of the string to vertical.