The drawing shows two crates that are connected by a steel (Young's modulus 2.0 x 1011 N/m2) wire that passes over a pulley. The unstretched length of the wire is 1.8 m, and its cross-sectional area is 1.4 x 10-5 m2. The pulley is frictionless and massless. When the crates are accelerating, determine the change in length of the wire. Ignore the mass of the wire.
The mass of the boxes are 3 kg and 5 kg
To determine the change in length of the wire, we can use Hooke's Law, which states that the change in length of an object is proportional to the applied force and inversely proportional to the object's stiffness.
1. Calculate the force applied to the wire:
- The force applied to the wire is the difference between the weight of the two crates since they are accelerating.
- The weight (force due to gravity) can be found using the equation: weight = mass * gravitational acceleration.
- The gravitational acceleration is approximately 9.8 m/s^2.
- The weight of the 3 kg crate is: weight1 = 3 kg * 9.8 m/s^2.
- The weight of the 5 kg crate is: weight2 = 5 kg * 9.8 m/s^2.
- The force applied to the wire is: force = weight2 - weight1.
2. Calculate the stress on the wire:
- Stress is defined as force divided by the cross-sectional area of the wire.
- Stress = force / area.
3. Determine the change in length of the wire using Hooke's Law:
- Hooke's Law states that stress is equal to the Young's modulus multiplied by the strain.
- Strain is defined as the change in length divided by the original length (unstretched length).
- Rearranging the equation: strain = stress / Young's modulus.
- The change in length of the wire is: change in length = strain * original length.
Let's calculate the change in length of the wire:
Given:
Young's modulus (E) = 2.0 × 10^11 N/m^2
Unstretched length (L) = 1.8 m
Cross-sectional area (A) = 1.4 × 10^-5 m^2
Mass of crate 1 (m1) = 3 kg
Mass of crate 2 (m2) = 5 kg
Step 1: Calculate the force applied to the wire.
weight1 = m1 * g = 3 kg * 9.8 m/s^2
weight2 = m2 * g = 5 kg * 9.8 m/s^2
force = weight2 - weight1
Step 2: Calculate the stress on the wire.
stress = force / A
Step 3: Calculate the change in length of the wire.
strain = stress / E
change in length = strain * L
By following these steps and plugging in the given values, you can calculate the change in length of the wire.