Two blocks with masses M1 = 8.0 kg and M2 = 5.4 kg are connected with a massless string over two massless and frictionless pulleys, as shown in the Figure. One end of the string is connected to M1 while the other end is fixed. What is the acceleration of mass M2?

Where is M2 in relation to the string?

3.4

To find the acceleration of mass M2, 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).

In this case, we will consider the forces acting on each block separately.

For M1:
The weight of M1 (W1) acts downward and is given by W1 = m1 * g (where g is the acceleration due to gravity).
The tension in the string (T) acts upward.
Applying Newton's Second Law to M1, we have:
m1 * a = T - m1 * g .......(Equation 1)

For M2:
The weight of M2 (W2) acts downward and is given by W2 = m2 * g (where g is the acceleration due to gravity).
The tension in the string (T) acts upward.
Applying Newton's Second Law to M2, we have:
m2 * a = m2 * g - T .......(Equation 2)

Since the blocks are connected by a string, the tension in the string is the same for both blocks. Therefore, we can equate the tension (T) in Equation 1 and Equation 2:
T - m1 * g = m2 * g - T

Simplifying this equation, we get:
2T = (m2 + m1) * g

Now, we need to solve for the tension (T). Dividing both sides of the equation by 2, we have:
T = [(m2 + m1) * g] / 2

Now that we have the tension in the string, we can substitute it back into either Equation 1 or Equation 2 to find the acceleration (a).

Let's use Equation 1:
m1 * a = T - m1 * g

Substituting the tension (T), we get:
m1 * a = [(m2 + m1) * g] / 2 - m1 * g

Simplifying this equation, we get:
a = [(m2 + m1) * g] / (2 * m1)

Now, we can substitute the given values:
m1 = 8.0 kg
m2 = 5.4 kg
g = 9.8 m/s^2

Hence, the acceleration of mass M2 is:
a = [(8.0 + 5.4) * 9.8] / (2 * 8.0) = 8.67 m/s^2