Two masses m1 = 2 kg and m2 = 4 kg are connected by a light string. Mass m2 is connected to mass M = 14 kg by a string that passes over an ideal pulley. Determine the following quantities when the system is released from rest.

(a) The acceleration of the masses

m/s2

(b) The tension in the string connecting M to m2

N

(b) The tension in the string connecting m2 to m1

N

0.2m/s^2

To determine the values, we can use Newton's second law of motion and the concepts of equilibrium. Let's break down the problem and solve it step by step.

Step 1: Identify the forces acting on each mass.

For mass m1 (2 kg):
- The tension in the string connecting m2 and m1, which we'll call T1, acts to the right.
- The force due to its mass, mg1 (where g is the acceleration due to gravity), acts downwards.

For mass m2 (4 kg):
- The tension in the string connecting m2 and m1 (which is also T1) acts to the left.
- The tension in the string connecting m2 and M, which we'll call T2, acts upwards.
- The force due to its mass, mg2, acts downwards.

For mass M (14 kg):
- The tension in the string connecting M to m2 (T2) acts downwards.
- The force due to its mass, Mg, acts downwards.

Step 2: Set up the equations of motion for each mass.

For mass m1:
Since m1 is only being acted upon by one force, we can write:
T1 - mg1 = m1a

For mass m2:
The net force acting on m2 is the difference between the two tensions, T1 and T2, minus the force due to its mass:
T2 - T1 - mg2 = m2a

For mass M:
The net force acting on M is simply the tension T2 minus the force due to its mass:
T2 - Mg = Ma

Step 3: Solve the equations simultaneously to obtain the values.

We have three equations (one for each mass) and three unknowns (a, T1, and T2). To simplify the calculation, we can eliminate one variable at a time.

Let's start by solving for a in the equation for m1:
T1 - mg1 = m1a --> T1 = m1a + mg1

Next, substitute this value into the equation for m2:
T2 - T1 - mg2 = m2a --> T2 - (m1a + mg1) - mg2 = m2a
T2 - m1a - mg1 - mg2 = m2a --> T2 - (m1a + mg1) - mg2 = m2a
T2 - mg1 - mg2 = (m2 + m1)a --> T2 = (m2 + m1)a + mg1 + mg2

Finally, substitute this value of T2 into the equation for M:
T2 - Mg = Ma --> ((m2 + m1)a + mg1 + mg2) - Mg = Ma
(m2 + m1)a + mg1 + mg2 - Mg = Ma
(m2 + m1)a = Mg - mg1 - mg2
a = (Mg - mg1 - mg2) / (m1 + m2)

Now that we have solved for a, we can substitute it back into the equations for T1 and T2 to find their values.

For T1:
T1 = m1a + mg1

For T2:
T2 = (m2 + m1)a + mg1 + mg2

Plug in the values of m1, m2, and M (2 kg, 4 kg, and 14 kg, respectively), and g (approx. 9.8 m/s^2) to calculate the specific values of a, T1, and T2.