The two barges shown here are coupled by a cable of negligible mass. The mass of the front barge is 2.22 ✕ 10^3 kg and the mass of the rear barge is 3.22 ✕ 10^3 kg. A tugboat pulls the front barge with a horizontal force of magnitude 34.0 ✕ 10^3 N, and the frictional forces of the water on the front and rear barges are 8.00 ✕ 10^3 N and 10.0 ✕ 10^3 N, respectively. (Enter the magnitudes.)
(a)Find the horizontal acceleration of the barges (in m/s2).
(b)Find the tension in the connecting cable (in N).
Correction: Please disregard the solution in part b.
(a) Well, it seems like these barges are really being pulled in different directions. It's like a tug of war, but on water! To find the horizontal acceleration, we first need to calculate the net force acting on the barges.
The net force can be determined by subtracting the frictional forces from the pulling force:
Net force = Pulling force - Frictional force on the front barge - Frictional force on the rear barge
Net force = (34.0 ✕ 10^3 N) - (8.00 ✕ 10^3 N) - (10.0 ✕ 10^3 N)
Now, since net force = mass × acceleration, we can rewrite the equation as:
(34.0 ✕ 10^3 N) - (8.00 ✕ 10^3 N) - (10.0 ✕ 10^3 N) = (2.22 ✕ 10^3 kg + 3.22 ✕ 10^3 kg) × acceleration
This equation allows us to solve for acceleration:
Acceleration = [(34.0 ✕ 10^3 N) - (8.00 ✕ 10^3 N) - (10.0 ✕ 10^3 N)] / (2.22 ✕ 10^3 kg + 3.22 ✕ 10^3 kg)
Calculating this, we find the horizontal acceleration of the barges to be (net force divided by total mass).
(b) Now, let's move on to the tension in the connecting cable. We can find it by analyzing the forces acting on it.
The tension in the cable is equal to the pulling force exerted by the tugboat, since the cable is being pulled directly by the tugboat itself.
Therefore, the tension in the connecting cable is 34.0 ✕ 10^3 N. Voila!
Remember, these are just the magnitudes of the answers, as requested. Stay afloat with your physics!
(a) To find the horizontal acceleration of the barges, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
The net force acting on the barges is the force applied by the tugboat minus the frictional forces:
Net force = Force applied by tugboat - Frictional force on front barge - Frictional force on rear barge
Net force = (34.0 ✕ 10^3 N) - (8.00 ✕ 10^3 N) - (10.0 ✕ 10^3 N)
Net force = 16.0 ✕ 10^3 N
The total mass of the barges is the sum of the mass of the front barge and the mass of the rear barge:
Total mass = Mass of front barge + Mass of rear barge
Total mass = 2.22 ✕ 10^3 kg + 3.22 ✕ 10^3 kg
Total mass = 5.44 ✕ 10^3 kg
Using Newton's second law, we can calculate the acceleration:
Net force = Total mass ✕ Acceleration
16.0 ✕ 10^3 N = 5.44 ✕ 10^3 kg ✕ Acceleration
Acceleration = (16.0 ✕ 10^3 N) / (5.44 ✕ 10^3 kg)
Acceleration = 2.94 m/s^2
Therefore, the horizontal acceleration of the barges is 2.94 m/s^2.
(b) To find the tension in the connecting cable, we can use Newton's second law again. Since the two barges are coupled by a cable of negligible mass, the tension in the cable will be the same as the force applied on the front barge by the tugboat:
Tension = Force applied by tugboat
Tension = 34.0 ✕ 10^3 N
Therefore, the tension in the connecting cable is 34.0 ✕ 10^3 N.
To find the horizontal acceleration of the barges (in m/s²), we need to use Newton's second law of motion. The equation is given by:
ΣF = m * a
where ΣF is the sum of all the forces acting on the system, m is the total mass of the system, and a is the acceleration.
In this case, the forces acting on the system are the tension in the cable (T), the force exerted by the tugboat (Ftugboat), and the frictional forces on the front and rear barges (Ffriction-front and Ffriction-rear respectively).
The equation can be written as:
T - Ftugboat - Ffriction-front - Ffriction-rear = (mass of front barge + mass of rear barge) * acceleration
Now let's calculate each force separately:
T - Ftugboat - Ffriction-front - Ffriction-rear = (2.22 × 10^3 kg + 3.22 × 10^3 kg) * acceleration
T - 34.0 × 10^3 N - 8.00 × 10^3 N - 10.0 × 10^3 N = (2.22 × 10^3 kg + 3.22 × 10^3 kg) * acceleration
T - 34.0 × 10^3 N - 8.00 × 10^3 N - 10.0 × 10^3 N = 5.44 × 10^3 kg * acceleration
T - 52.0 × 10^3 N = 5.44 × 10^3 kg * acceleration
T = 5.44 × 10^3 kg * acceleration + 52.0 × 10^3 N
Now, to calculate the tension in the cable (T), we need to solve for acceleration. Rearranging the equation, we get:
acceleration = (T - 52.0 × 10^3 N) / (5.44 × 10^3 kg)
Now we can substitute the given values into the equation and solve for acceleration:
Acceleration = (T - 52.0 × 10^3 N) / (5.44 × 10^3 kg)
To find the tension in the connecting cable (T), we can substitute this acceleration value back into any of the force equations:
T = 5.44 × 10^3 kg * acceleration + 52.0 × 10^3 N
Substituting the calculated acceleration, we can solve for T.