By accident, a large plate is dropped and breaks into three pieces. The pieces fly apart parallel to the floor. As the plate falls, its momentum has only a vertical component, and no component parallel to the floor. After the collision, the component of the total momentum parallel to the floor must remain zero, since the net external force acting on the plate has no component parallel to the floor. Using the data shown in the drawing find (a) the mass of piece 1 and (b) the mass of piece 2.
To solve this problem, we need to use the principles of conservation of momentum and the fact that the momentum component parallel to the floor remains zero throughout the motion.
Let's denote the masses of piece 1, piece 2, and piece 3 as m1, m2, and m3, respectively. The initial momentum of the plate before it breaks can be calculated from the equation:
Initial momentum = Sum of momentum of individual pieces
Given that the initial momentum has only a vertical component and no component parallel to the floor, we can write:
Initial momentum = m1 * 0 + m2 * 0 + m3 * 0
Since any term multiplied by 0 is 0, the equation simplifies to:
Initial momentum = 0
This equation tells us that the total momentum before the collision is zero.
Now, let's analyze the momentum after the collision. After breaking, the plate is split into three pieces, but still has no momentum parallel to the floor. Therefore, the vertical components of the momenta of the three pieces must add up to zero.
From the drawing, we see that piece 1 has momentum upward, while pieces 2 and 3 have downward momentum. Using the positive direction as upward, we can write:
m1v1 - m2v2 - m3v3 = 0
Here, v1, v2, and v3 represent the velocities of pieces 1, 2, and 3, respectively, after the collision.
Since we don't have values for any of the variables, it's not possible to find the actual masses of the pieces. However, we can determine the ratio between the masses.
Let's represent the ratio of m2 to m1 as r (m2/m1 = r), and the ratio of m3 to m1 as s (m3/m1 = s). By substituting these ratios into the equation above, we obtain:
m1v1 - (r * m1)v2 - (s * m1)v3 = 0
Dividing the equation through by m1:
v1 - r * v2 - s * v3 = 0
This equation relates the velocities of the three pieces after the collision.
Unfortunately, without more information or numerical values, we are unable to calculate the masses m1 and m2 specifically, but we can determine their ratio by comparing their respective velocities using the equation above.