a bullet of mass 20g traveling horizontally at 100m/s embedded itself in the center of block of wood mass 1kg which is suspended by a light vertical string of length 1m. calculate the maximum inclination of the string to the vertical
Conservation of linear momentum applies to the embedding process. Use that to calculate the velocity at impact, before swinging begins.
Then use that velocity and conservation of energy to calculate how high it swings.
V = sqrt (2 g H) and the height that is reaches is
H = L (1 - cos A)
Use that to solve for the maximum incination angle A
To calculate the maximum inclination of the string to the vertical, we need to consider the conservation of angular momentum.
1. First, we will find the initial angular momentum of the bullet-wood system before the collision.
The initial angular momentum (L_i) is given by the product of moment of inertia (I) and angular velocity (ω).
Since the system is moving horizontally and the bullet is attached to the center of the block, the initial angular momentum is zero.
L_i = 0
2. After the bullet embeds itself in the wood, the system starts rotating about the center of mass.
The final angular momentum (L_f) can be calculated from the moment of inertia of the block and angular velocity after the collision.
L_f = I_f * ω_f
3. To calculate the final angular velocity, we can use the principle of conservation of linear momentum.
The total linear momentum before the collision is equal to the linear momentum after the collision.
m_bullet * v_bullet = (m_bullet + m_wood) * v_f
Here, m_bullet is the mass of the bullet and v_bullet is its initial velocity.
m_wood is the mass of the wood block and v_f is the final velocity of the bullet-wood system.
4. Since the bullet embedded itself in the center of the wood, the final velocity (v_f) is equal to the final angular velocity (ω_f) multiplied by the radius of the wood block.
v_f = r * ω_f
5. Substituting the values from steps 3 and 4, we have:
m_bullet * v_bullet = (m_bullet + m_wood) * r * ω_f
6. Rearranging the equation, we can solve for ω_f:
ω_f = (m_bullet * v_bullet) / ((m_bullet + m_wood) * r)
7. The maximum inclination of the string occurs when the angular velocity is maximum, which happens when ω_f takes its maximum value.
Maximum ω_f = (m_bullet * v_bullet) / ((m_bullet + m_wood) * r)
8. Finally, we can calculate the maximum inclination angle (θ) using the formula:
θ = arctan(ω_f / √(g / L))
Here, g is the acceleration due to gravity and L is the length of the string.
By substituting the given values into the equations, you can find the maximum inclination of the string to the vertical.
To calculate the maximum inclination angle of the string, we need to consider the conservation of energy.
First, let's calculate the initial kinetic energy of the bullet:
Initial kinetic energy (KE1) = (1/2) * mass of the bullet * (velocity of the bullet)^2
Given:
Mass of the bullet = 20 g = 0.02 kg
Velocity of the bullet = 100 m/s
KE1 = (1/2) * 0.02 kg * (100 m/s)^2 = 100 J
Next, let's calculate the maximum potential energy of the block of wood and bullet system when it reaches its maximum inclination:
Maximum potential energy (PE) = mass of the block * g * height
Given:
Mass of the block of wood = 1 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Height = length of the string = 1 m
PE = 1 kg * 9.8 m/s^2 * 1 m = 9.8 J
Since energy is conserved, the initial kinetic energy of the bullet (KE1) is equal to the maximum potential energy of the system (PE) at the maximum inclination angle.
Therefore, the maximum inclination angle (θ) can be calculated by equating the two energies:
KE1 = PE
100 J = 9.8 J
Dividing both sides by 9.8 J:
100 J / 9.8 J = 1
Thus, the maximum inclination angle (θ) is approximately 1 radian.