"By means of a rope, whose mass is negligible, two blocks (mass: 11kg and 44 kg) are suspended over a pulley. The pulley can be treated as a uniform solid cylindrical disk. The downward acceleration of the 44 kg block is observed to be exactly one half the acceleration due to gravity. Noting that the tension in the rope is not the same on each side of the pulley, find the mass of the pulley."

The answer of 22 kilograms is in the back of the book, but I need someone to point me in the right direction as to the steps I need to take to arrive at this solution.

To find the mass of the pulley, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = ma).

Let's break down the problem step by step:

1. First, determine the gravitational force acting on each block. The weight or gravitational force (Fg) is calculated by multiplying the mass (m) of an object by the acceleration due to gravity (g = 9.8 m/s^2).

For the 11 kg block:
Fg1 = m1 * g
= 11 kg * 9.8 m/s^2

For the 44 kg block:
Fg2 = m2 * g
= 44 kg * 9.8 m/s^2

2. Since the 44 kg block has an acceleration that is half of the acceleration due to gravity, we can write the equation for its net force in terms of a fraction:

F2 = m2 * (g/2)

3. Now, let's consider the tension in the rope on each side of the pulley. Since the rope is assumed to have a negligible mass, the tension on both sides of the pulley has the same magnitude, but opposite directions.

Let T1 be the tension on the side of the 11 kg block, and T2 be the tension on the side of the 44 kg block.

4. The net force acting on the 11 kg block is given by:
Fnet1 = T1 - Fg1

5. For the 44 kg block, the net force can be expressed as:
Fnet2 = T2 - Fg2

6. The relationship between the accelerations of the two blocks and the pulley can be determined by considering the fact that they are connected by the same rope. Since the rope is inextensible, the displacements of the blocks will be equal in magnitude. Hence, the magnitudes of the acceleration of each block can be related to the acceleration (a) of the pulley.

The 11 kg block:
a1 = -a

The 44 kg block:
a2 = 2a

Note that the negative sign for a1 indicates that it is downward, while the positive sign for a2 corresponds to upward acceleration.

7. We can now relate the tensions and masses using Newton's second law.

For the 11 kg block:
Fnet1 = m1 * a1
T1 - Fg1 = 11 kg * (-a)

For the 44 kg block:
Fnet2 = m2 * a2
T2 - Fg2 = 44 kg * (2a)

8. We can rearrange the equations and solve for T1 and T2:

T1 = Fg1 - 11 kg * (-a)
T2 = Fg2 + 44 kg * (2a)

9. Since the tension on both sides of the pulley is not the same, we can also write the relationship between T1 and T2 using the pulley's acceleration a:

T2 = T1 - I * alpha

Where I is the moment of inertia of the pulley and alpha is the angular acceleration. For a solid cylindrical disk, the moment of inertia is given by I = (1/2) * m_pulley * r^2, where m_pulley is the mass of the pulley and r is the radius.

10. Combining the equations from steps 8 and 9, we have:

Fg2 + 44 kg * (2a) = Fg1 - 11 kg * (-a) - I * alpha

11. Since the pulley is not accelerating in rotation (we assume the rope does not slip on the pulley), the angular acceleration alpha is related to the linear acceleration a and the radius r by the expression a = r * alpha.

12. With a = r * alpha, we can rewrite the equation from step 10:

Fg2 + 44 kg * (2a) = Fg1 - 11 kg * (-a) - (1/2) * m_pulley * r^2 * a / r

13. Simplifying the equation further:

Fg2 + 88 kg * a = Fg1 + 11 kg * a - (1/2) * m_pulley * r * a

14. Now we know that the weight of an object is equal to its mass multiplied by the acceleration due to gravity. So we can substitute Fg1 and Fg2:

m2 * g + 88 kg * a = m1 * g + 11 kg * a - (1/2) * m_pulley * r * a

15. Rearrange the equation:

m1 * g + (1/2) * m_pulley * r * a + 77 kg * a - m2 * g = 0

16. Now we can solve for the acceleration a:

a = (m2 * g - m1 * g) / ((1/2) * m_pulley * r + 77 kg)

17. We are given that the acceleration a equals half the acceleration due to gravity, so:

a = (1/2) * g
= 1/2 * 9.8 m/s^2

18. Substitute the known values into the acceleration equation:

1/2 * 9.8 m/s^2 = (m2 * 9.8 m/s^2 - m1 * 9.8 m/s^2) / ((1/2) * m_pulley * r + 77 kg)

19. Simplify the equation:

1 = (m2 - m1) / ((1/2) * m_pulley * r + 77 kg)

20. Solve for the mass of the pulley, m_pulley:

(m2 - m1) / ((1/2) * m_pulley * r + 77 kg) = 1

(m2 - m1) = (1/2) * m_pulley * r + 77 kg

m_pulley = ((m2 - m1) - 77 kg) / ((1/2) * r)

21. Finally, substitute the mass values given in the problem:

m_pulley = ((44 kg - 11 kg) - 77 kg) / ((1/2) * r)

Simplifying further yields:

m_pulley = 22 kg

Therefore, the mass of the pulley is 22 kg.

To find the mass of the pulley, we can use Newton's second law of motion and analyze the forces acting on each block.

Step 1: Identify the forces acting on the 11 kg block:
- Tension force (T) acting upward
- Weight force (mg) acting downward

Step 2: Write the equation of motion for the 11 kg block using Newton's second law:
ma = T - mg, where 'a' is the acceleration

Step 3: Identify the forces acting on the 44 kg block:
- Tension force (T) acting downward
- Weight force (mg) acting downward

Step 4: Write the equation of motion for the 44 kg block using Newton's second law:
ma = mg - T, where 'a/2' is the acceleration

Step 5: Relate the accelerations of both blocks:
The downward acceleration of the 44 kg block is observed to be exactly one-half the acceleration due to gravity. This means that a/2 = g/2, where 'g' is the acceleration due to gravity (approximately 9.8 m/s^2).

Step 6: Solve the equations simultaneously:
Using the known mass values (m₁ = 11 kg, m₂ = 44 kg) and acceleration values (a/2 = g/2), substitute the values into the equations:

For the 11 kg block: 11a = T - (11)(9.8)
For the 44 kg block: 44(a/2) = (44)(9.8) - T

Simplify the equations:

11a - T = -107.8
22a - 2T = 431.2

Multiply the first equation by 2:

22a - 2T = -215.6

Now, equate the two equations and solve for 'a':

22a - 2T = 22a - 107.8
-215.6 = -107.8

This means that -2T = -107.8, and thus T = 53.9 N.

Step 7: Find the mass of the pulley:
The tension force on the side with the 11 kg block is 53.9 N. The tension force on the side with the 44 kg block is T (unknown), so we need to determine its value.

Using the equation: 44(a/2) = (44)(9.8) - T, substitute the values into the equation:

44(a/2) = (44)(9.8) - 53.9

Solve for 'a':

22a = 431.2
a = 431.2/22
a ≈ 19.6 m/s²

Substitute the value of 'a' into the first equation:

11a = T - (11)(9.8)
(11)(19.6) = T - (11)(9.8)
215.6 = T - 107.8
T = 323.4 N

Since the tension force on both sides of the pulley is not the same, the difference in tension is due to the mass of the pulley. Therefore, the mass of the pulley can be determined by using the equation:

Tension difference = mass of the pulley × acceleration
323.4 - 53.9 = mass of the pulley × 19.6

Simplify:

269.5 = mass of the pulley × 19.6

Divide both sides by 19.6:

mass of the pulley ≈ 269.5 / 19.6
mass of the pulley ≈ 13.75 kg

Therefore, the mass of the pulley is approximately 13.75 kg.