A 12 g rifle bullet traveling 220 m/s buries itself in a 3.3 kg pendulum hanging on a 2.6 m long string, which makes the pendulum swing upward in an arc. Determine the horizontal component of the pendulum's maximum displacement.
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To determine the horizontal component of the pendulum's maximum displacement, we can use the principle of conservation of linear momentum.
The initial linear momentum is given by the bullet's mass (m1) times its velocity (v1). The final linear momentum is given by the pendulum's mass (m2) times the velocity (v2) of the bullet and pendulum together.
Since the bullet buries itself in the pendulum, we can assume that the bullet and the pendulum move together as a single unit after the collision. Therefore, the final velocity (v2) is the same for both the bullet and the pendulum.
Using the conservation of linear momentum, we can write:
(m1)v1 = (m1 + m2)v2
Rearranging the equation to solve for v2:
v2 = (m1 * v1) / (m1 + m2)
Now, the horizontal component of the pendulum's maximum displacement can be determined using the equation for maximum displacement of a simple pendulum:
x_max = L * sin(theta)
where L is the length of the string and theta is the maximum angle of displacement of the pendulum.
Given:
m1 = 0.012 kg (mass of the bullet)
v1 = 220 m/s (velocity of the bullet)
m2 = 3.3 kg (mass of the pendulum)
L = 2.6 m (length of the string)
First, we calculate v2:
v2 = (0.012 kg * 220 m/s) / (0.012 kg + 3.3 kg)
= (2.64 kg⋅m/s) / (3.312 kg)
≈ 0.797 m/s
Now, we need to find the maximum angle of displacement (theta) of the pendulum. To do this, we can use the conservation of mechanical energy.
The initial kinetic energy of the bullet is given by:
KE1 = 0.5 * m1 * v1^2
The final kinetic energy of the bullet and the pendulum together is given by:
KE2 = 0.5 * (m1 + m2) * v2^2
Since there is no external force doing work on the system, the initial kinetic energy of the bullet must equal the final kinetic energy of the bullet and pendulum.
Setting KE1 equal to KE2:
0.5 * m1 * v1^2 = 0.5 * (m1 + m2) * v2^2
Simplifying the equation and solving for v2:
v2^2 = (m1 * v1^2) / (m1 + m2)
Inserting the given values:
v2^2 = (0.012 kg * (220 m/s)^2) / (0.012 kg + 3.3 kg)
≈ 107.42 m^2/s^2
Now, we can determine the maximum angle of displacement (theta) using the equation for the conservation of mechanical energy:
v2^2 = 2 * g * L * (1 - cos(theta))
where g is the acceleration due to gravity.
Rearranging the equation for theta:
theta = acos(1 - (v2^2 / (2 * g * L)))
Given:
g ≈ 9.8 m/s^2
Plugging in the values:
theta = acos(1 - (107.42 m^2/s^2 / (2 * 9.8 m/s^2 * 2.6 m)))
≈ acos(1 - 0.216)
≈ 1.336 radians
Finally, we can calculate the horizontal component of the pendulum's maximum displacement:
x_max = L * sin(theta)
= 2.6 m * sin(1.336 radians)
≈ 3.023 m
Therefore, the horizontal component of the pendulum's maximum displacement is approximately 3.023 m.