Determining the Acceleration of a Mass System Using Newton’s Laws

Consider a hanging mass m1 0.500 kg

attached to a frictionless table mass m2 1.50 kg by a light string that runs over a frictionless pulley. This configuration can be analyzed using Newton’s laws of motion, specifically the second law, to determine the acceleration of the masses.

The Forces Acting on the System

To determine the acceleration, we start by identifying the forces acting on the system.

Force on Mass (m_1)

The tension in the string, T, is the only force acting on the hanging mass, and it generates the acceleration (a_1).

F1 T (m_1 a_1)

Force on Mass (m_2)

The tension in the string, T, minus the gravitational force, m2g, generates the acceleration (a_2) in the opposite direction to the tension.

F2 T - (m_2 g) ( -m_2 a_2)

Analysis Using Newton's Second Law

Since the masses are attached to the same string, their accelerations are equal in magnitude but opposite in direction, i.e., (a_1 a_2) and thus, the net force is applied only to the mass on the table.

Considering the second law, we can write:

For Mass (m_1)

F1 (m_1 a T)

For Mass (m_2)

F2 (T - m_2 g -m_2 a)

Substituting and Solving for Acceleration

By substituting (T m_1 a) into the equation for (m_2), we get:

T - (m_2 g -m_2 a)

Substitute T with (m_1 a):

(m_1 a - m_2 g -m_2 a)

Rearranging gives:

(m_1 a m_2 a m_2 g)

((m_1 m_2) a m_2 g)

(a frac{m_2}{m_1 m_2} g)

Calculating the Acceleration

Using the given values:

(m_1 0.500 , text{kg}) and (m_2 1.50 , text{kg})

(g 9.81 , text{m/s}^2)

(a frac{1.50}{0.500 1.50} times 9.81 frac{1.50}{2.00} times 9.81 0.75 times 9.81 7.36 , frac{text{m}}{text{s}^2})

Alternative Approach with Free-Body Diagrams

Alternatively, we solve this problem by creating free-body diagrams and applying the second law equations. This approach ensures a thorough understanding of the forces at play.

Free-Body Diagram for m2 on the Table

The net force on the mass on the table is simply the gravitational force contributing to the acceleration:

(F -m_2 g -m_2 a)

Since both masses are moving together, their acceleration is the same. Solving for (a) gives:

(a frac{m_2}{m_1 m_2} g)

This yields the same result.

Conclusion

Using Newton’s second law and free-body diagrams, we can accurately determine the magnitude of the acceleration of a mass system. The key takeaway is the importance of understanding and applying the principles of physics to solve real-world problems.