To analyze the motion of the stick after being struck by the puck, we need to apply the principles of linear momentum and angular momentum. The stick's response will involve both translational and rotational motion due to the nature of the impact. Let's break this down step by step.
Understanding the System
The uniform stick has a mass of 4.42 kg and a length of 1.23 m. When the puck strikes it with an impulse of 12.8 N·s at a distance of 46.4 cm (or 0.464 m) from the center, we can expect two main outcomes:
- The stick will start to translate (move) due to the horizontal force applied by the puck.
- The stick will also begin to rotate about its center of mass because the force is applied off-center.
Calculating Translational Motion
First, let's determine the stick's linear acceleration after the impulse is applied. The impulse-momentum theorem states that the impulse applied to an object equals its change in momentum.
The initial momentum of the stick is zero since it is at rest. The impulse given to the stick is 12.8 N·s, which will be its final momentum:
Final Momentum (p) = Impulse = 12.8 N·s
Now, we can find the final velocity (v) of the stick using the formula:
p = m * v
Rearranging gives:
v = p / m
Substituting the values:
v = 12.8 N·s / 4.42 kg ≈ 2.9 m/s
Determining Rotational Motion
Next, we need to analyze the rotational motion. The torque (τ) caused by the impulsive force needs to be calculated. Torque is defined as:
τ = r × F
Where:
- r = distance from the center of mass to the point of force application = 0.464 m
- F = average force applied (which can be derived from impulse) = Impulse / time; however, we can directly calculate angular momentum without knowing time.
For this case, since we have the impulse, we can directly calculate the torque around the center of mass:
Since the impulse is applied horizontally, we can treat it as causing a torque about the center:
τ = 0.464 m × (12.8 N / Δt)
Next, we calculate the moment of inertia (I) of the stick about its center of mass:
I = (1/12) * m * L²
Here, L is the length of the stick:
I = (1/12) * 4.42 kg * (1.23 m)² ≈ 0.431 kg·m²
Finding Angular Velocity
The change in angular momentum (ΔL) is equal to the torque multiplied by the time interval:
ΔL = τ × Δt
Using conservation of angular momentum:
Initial angular momentum is zero, thus:
τ = I * ω
Where ω is the angular velocity. Rearranging gives:
ω = τ / I
Using our earlier torque calculation:
We can estimate the angular velocity if we consider the average force over a very small time interval, but without loss of generality, we can summarize the motion qualitatively.
Final Motion of the Stick
After the puck strikes the stick, it will:
- Translate with a velocity of approximately 2.9 m/s.
- Rotate about its center of mass with a certain angular velocity, which can be calculated more precisely if the exact time of contact is known.
This combined motion results in the stick moving forward while simultaneously spinning due to the off-center impact point. Such a scenario is a classic example of how forces can lead to both translation and rotation in rigid bodies.
