A uniform thin rod, 7.0 cm long with a mass of 40.0 g, lies on a frictionless horizontal table. It is struck with a horizontal impulse, at right angles to its length, at a point 2.0 cm from one end. If the impulse is 8.5 mN-s, describe the resulting motion of the stick.
To describe the resulting motion of the stick, we need to analyze the moment of inertia and the conservation of angular momentum.
Step 1: Determine the moment of inertia of the rod.
The moment of inertia of a thin rod about an axis passing through its center and perpendicular to its length is given by the formula: I = (1/12) * m * L^2, where m is the mass and L is the length of the rod.
Given:
Mass of the rod (m) = 40.0 g = 0.040 kg
Length of the rod (L) = 7.0 cm = 0.07 m
Using the formula, I = (1/12) * m * L^2, we can calculate the moment of inertia:
I = (1/12) * 0.040 kg * (0.07 m)^2
I = 0.0000168 kg*m^2
Step 2: Apply the conservation of angular momentum.
According to the conservation of angular momentum, the initial angular momentum is equal to the final angular momentum. The initial angular momentum is zero since the rod is at rest. The final angular momentum can be calculated using the formula: L = I * ω, where ω is the angular velocity.
Given:
Impulse (J) = 8.5 mN-s = 8.5 x 10^-3 N-s
Distance from the point of impact to the center of mass (r) = 2.0 cm = 0.02 m
The impulse J can be related to angular momentum using the equation J = ΔL = I * Δω, where ΔL is the change in angular momentum and Δω is the change in angular velocity.
Since the rod is struck at right angles to its length, the impulse is directly related to the angular velocity change. So J = ΔL = I * Δω.
Δω can be found by rearranging the equation: Δω = ΔL / I
Δω = J / I
Δω = (8.5 x 10^-3 N-s) / (0.0000168 kg*m^2)
Δω = 505.95 rad/s (approximately)
Step 3: Determine the resulting motion.
Since the rod is struck at a point off-center, it will experience angular acceleration and start rotating about its center of mass.
The resulting motion of the stick will be rotational motion. The point of impact will start to rotate away from the direction of the impulse because an angular velocity is imparted to the rod. The rotation will continue until the rod comes to rest or until another external force acts on it.
In summary, the stick will rotate about its center of mass due to the applied impulse, with the point of impact moving away from the direction of the impulse.