Two crates, one with mass 4.00 kg and the other with mass 6.00 kg, sit on the frictionless surface of a frozen pond, connected by a light rope. A woman wearing golf shoes (so she can traction on the ice) pulls horizontally on the 6.00 kg crate with a force F that gives the crate an acceleration of 2.50 m/s2. a) What is the magnitude of the force F? b) What is the tension in the rope connecting the two crates?
I assume the rope starts taut
F = m a = (4 + 6) 2.5 = 25 N to pull both
F = 4 *2.5 = 10 N to pull just the 4 kg one
a) Well, well, well! Looks like someone's got some muscles! To find the magnitude of the force F, we can use Newton's second law, which states that force is equal to mass times acceleration. Here, we have the mass of the 6.00 kg crate and the acceleration provided (2.50 m/s²). So F = (6.00 kg)(2.50 m/s²). Calculate that, my friend, and you'll have the magnitude of force F.
b) Ah, the tension in the rope. We need to find that, eh? Alright, listen carefully. Since the two crates are connected by a rope, the tension in the rope is the same for both crates. Mr. Newton would tell you that the net force acting on the 4.00 kg crate is equal to the mass of the crate times its acceleration. And since there are no horizontal external forces acting on it (frictionless surface, remember?), the net force equals the tension in the rope. So, tension = (4.00 kg)(2.50 m/s²).
Now, give those numbers a good workout with the calculator, and you'll have your answers!
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration (F = ma). We will apply this law to each crate separately and then solve for the unknowns.
a) To find the magnitude of the force F, we will use the mass and acceleration of the 6.00 kg crate.
Given:
Mass of the 6.00 kg crate (m) = 6.00 kg
Acceleration of the 6.00 kg crate (a) = 2.50 m/s^2
Using Newton's second law of motion (F = ma), we can calculate the force F:
F = m * a
F = 6.00 kg * 2.50 m/s^2
F = 15.00 N
Therefore, the magnitude of the force F is 15.00 N.
b) To find the tension in the rope connecting the two crates, we need to consider the forces acting on the 4.00 kg crate. Since the surface is frictionless, the only forces acting on the 4.00 kg crate are the tension force (T) applied by the rope and the force exerted by the woman on the 4.00 kg crate (F).
Let's denote the tension force in the rope as T and the force exerted by the woman as F.
Since the crates are connected by a rope, the magnitudes of the tension force and the force exerted by the woman are equal. Therefore, T = F.
Using Newton's second law for the 4.00 kg crate (F = ma), we can rewrite F as F = 4.00 kg * 2.50 m/s^2 = 10.00 N.
Therefore, the tension in the rope connecting the two crates is also 10.00 N.
To find the magnitude of the force F, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.
a) Using this formula, we have:
Force (F) = mass (m) × acceleration (a)
Given:
mass (m) = 6.00 kg
acceleration (a) = 2.50 m/s^2
Substituting these values into the formula, we get:
F = 6.00 kg × 2.50 m/s^2
Multiplying these values:
F = 15.00 N
Therefore, the magnitude of the force F is 15.00 Newtons.
b) To find the tension in the rope connecting the two crates, we need to consider the system as a whole.
Since the crates are connected by a light rope, they experience the same tension (T) in the rope.
The force F applied by the woman pulling the 6.00 kg crate creates tension in the rope, which in turn pulls the 4.00 kg crate.
We know that the mass of the 4.00 kg crate is less than the 6.00 kg crate. Therefore, the tension in the rope is responsible for accelerating both crates.
Using Newton's second law, we can relate the tension with the acceleration of the entire system.
We can calculate the net acceleration of the system by comparing the mass of the two crates:
Net acceleration = m1 × a1 / (m1 + m2)
Where:
m1 = mass of the 4.00 kg crate
a1 = acceleration
Given:
m1 = 4.00 kg
a1 = 2.50 m/s^2
Substituting these values into the formula, we get:
Net acceleration = (4.00 kg × 2.50 m/s^2) / (4.00 kg + 6.00 kg)
Calculating:
Net acceleration = 10.00 kg·m/s^2 / 10.00 kg
Net acceleration = 1.00 m/s^2
Since both crates are accelerating with the same net acceleration, the tension in the rope connecting them is equal to the force required to accelerate the net mass of the system.
Using the same formula as before (F = m · a), we have:
Tension (T) = (m1 + m2) × net acceleration
Substituting the values we have:
T = (4.00 kg + 6.00 kg) × 1.00 m/s^2
Calculating:
T = 10.00 kg × 1.00 m/s^2
T = 10.00 N
Therefore, the tension in the rope connecting the two crates is 10.00 Newtons.