Question

An elevator contains two masses: \(M_{1}=2.00 \mathrm{~kg}\) is attached by a string (string 1 ) to the ceiling of the elevator, and \(M_{2}=4.00 \mathrm{~kg}\) is attached by a similar string (string 2) to the bottom of mass 1 . a) Find the tension in string \(1\left(T_{1}\right)\) if the elevator is moving upward at a constant velocity of \(v=3.00 \mathrm{~m} / \mathrm{s}\) b) Find \(T_{1}\) if the elevator is accelerating upward with an acceleration of \(a=3.00 \mathrm{~m} / \mathrm{s}^{2}\)

Short answer

Answer: (a) The tension in string 1, \(T_{1}\), when the elevator is moving upward with a constant velocity is \(19.62\,\text{N}\). (b) The tension in string 1, \(T_{1}\), when the elevator is accelerating upward at \(3.00\,\text{m/s}^2\) is \(25.62\,\text{N}\).

Step-by-step solution

Draw a free body diagram for both masses M1 and M2

For M1, upward force on it is \(T_{1}\) and downward force is its weight, \(M_{1}g\). Likewise, for \(M_{2}\), upward force is \(T_{2}\) and downward force is its weight, \(M_{2}g\).

Apply Newton's second law to the system

Since the elevator is moving with constant velocity (zero acceleration), the net force acting on each mass is zero. Make use of Newton's second law for both masses as follows: For M1: \(T_{1} - M_{1}g = 0\) For M2: \(T_{1} - T_{2} - M_{2}g = 0\)

Solve the equations for T1

As we are interested in the tension in string 1 (\(T_{1}\)), we can solve the first equation for \(T_1\). \(T_1 = M_{1}g = (2.00\,\text{kg})(9.81\,\text{m/s}^2) = 19.62\,\text{N}\) Answer (a): The tension in string 1, \(T_{1}\), when the elevator is moving upward with a constant velocity is \(19.62\,\text{N}\). b) Upward acceleration (\(a=3.00 \mathrm{~m} / \mathrm{s}^{2}\))

Apply Newton's second law to the system

Now, the elevator is accelerating upward. Apply Newton's second law for both masses as follows: For M1: \(T_{1} - M_{1}(g+a) = 0\) For M2: \(T_{1} - T_{2} - M_{2}(g+a) = 0\)

Solve the equations for T1

We can solve the first equation for \(T_1\). \(T_{1} = M_{1}(g+a) = (2.00\,\text{kg})(9.81\,\text{m/s}^2 + 3.00\,\text{m/s}^2) = 25.62\,\text{N}\) Answer (b): The tension in string 1, \(T_{1}\), when the elevator is accelerating upward at \(3.00\,\text{m/s}^2\) is \(25.62\,\text{N}\).

Key concepts

Newton's second law

Newton's second law is fundamental in understanding how objects move and interact with forces. In essence, it tells us that the force (\textbf{F}) acting on an object is equal to the mass (\textbf{m}) of that object multiplied by the acceleration (\textbf{a}) it experiences. The formula is expressed as \( F = m \cdot a \).

When we analyze scenarios with constant velocity, the net force is zero \( F_{net} = 0 \), implying that acceleration is also zero. This is because, according to Newton's second law, if there's no net force, there's no acceleration. Conversely, when there’s a net force, the object will accelerate in the direction of that force. This law serves as the bridge between an object's motion and the forces acting upon it.

Free body diagram

A free body diagram is a visual tool used in physics to illustrate the forces acting on an object. This helps in analyzing the motions of objects and solving problems involving forces. To sketch one, you represent the object by a dot or a simple shape, and then draw arrows emanating from it to represent each force. The size and direction of these arrows reflect the magnitude and direction of the forces.

In the context of our elevator problem, we drew the free body diagram to understand the forces acting on the masses. By doing this, we were able to set up equations that help us calculate the tension in the strings, using Newton's second law. For complex systems, free body diagrams are essential for simplifying and breaking down force interactions.

Constant velocity

Constant velocity implies that an object is moving at a consistent speed and direction. When an object travels with constant velocity, it means it’s neither speeding up nor slowing down—it’s in a state of uniform motion. According to Newton's first law, also known as the law of inertia, an object in motion at a constant velocity will stay in motion at that velocity unless acted upon by a net external force.

In the elevator example with constant velocity, it indicates that there are no unbalanced forces acting on the system and the net force is zero. This information allows us to conclude that the acceleration is zero, which simplifies our calculations and understanding of the system's dynamics.

Acceleration

Acceleration is the rate at which the velocity of an object changes with time. It is a vector quantity, meaning it has both magnitude and direction. Acceleration can be due to changes in speed, changes in direction, or both. The standard unit of acceleration is meters per second squared (\(\text{m/s}^2\)).

When applying Newton's second law, if an object is accelerating, there must be a net force acting on it. In the scenario where the elevator is accelerating upward, we calculate this force to determine the new tension in the strings, illustrating the direct relationship between unbalanced forces and the acceleration they generate. This is crucial for solving problems involving dynamics and forces.