A 4.0 g bullet is shot with an initial velocity of 300.0 m/s into a board (I = 19.0 kg m2). The board is held still by a frictionless pivot a third away from its bottom end, as shown below. Assume that you can treat the board as a uniform rod.
1)
What is the initial angular momentum of the bullet with respect to the axle?
2)
What is the distance between the axis of rotation and the center of mass of the board?
3)
If the board is initially uniform, what is its mass?
4)
Once the bullet enters the board, what is its contribution to the moment of inertia?
5)
What is the angular velocity of the board + bullet system after the collision?
To answer these questions, we can use the principles of rotational dynamics, specifically the conservation of angular momentum and the concept of moment of inertia.
1) The initial angular momentum of the bullet with respect to the axle can be calculated using the formula L = r*m*v, where L is the angular momentum, r is the distance from the axle, m is the mass of the bullet, and v is the velocity of the bullet. In this case, the bullet is shot into the board, so its initial distance from the axle is the third of the length of the board. The mass of the bullet is given as 4.0 g (or 0.004 kg), and the initial velocity is 300.0 m/s. Therefore, the initial angular momentum is L = (1/3) * 0.004 kg * 300.0 m/s.
2) The distance between the axis of rotation and the center of mass of the board can be determined by dividing the length of the board by 2, since the board is symmetric and the pivot point is a third away from the bottom of the board.
3) If the board is initially uniform, its mass can be calculated using the formula for the moment of inertia of a uniform rod, I = (1/3) * m * L^2, where I is the moment of inertia, m is the mass of the object, and L is the length of the object. Rearranging the formula, we find that m = (3 * I) / (L^2). In this case, the moment of inertia of the board is given as 19.0 kg*m^2, and the length of the board is needed to calculate the mass.
4) Once the bullet enters the board, its contribution to the moment of inertia can be calculated using the formula for the moment of inertia of a point mass, I_bullet = m_bullet * r^2, where I_bullet is the moment of inertia of the bullet, m_bullet is the mass of the bullet, and r is the distance of the bullet from the axis of rotation. The mass of the bullet is given as 4.0 g (or 0.004 kg), and the distance from the axis of rotation can be determined by considering the position of the bullet within the board.
5) To find the angular velocity of the board + bullet system after the collision, we can apply the principle of conservation of angular momentum. The initial angular momentum of the system is equal to the final angular momentum, assuming no external torques act on the system. We can calculate the initial angular momentum of the system by summing the angular momenta of the bullet and the board, and then divide by the total moment of inertia of the system.
It's important to note that in answering these questions, the actual calculations require the specific values of the distances and lengths, as well as the specific position of the bullet within the board.