The figure below shows two objects with masses m and M which are connected with a taut string running over a pulley. The pulley rotates without friction. The two masses are given as M = 393 g and m = 181 g. A second taut string connects the heavier object vertically from the ceiling. Assuming that both strings and the pulley are mass less, calculate the tension in the string running over the pulley.


Express the result in the unit N and to three significant figures

M=0.393kg

m=0.181kg

Draw the FBD, you will see that

T1+T2-W(of M)=0 Eq 1
T2-W(of m)=0 Eq 2

(T2 is what we need)

now, Eq1-Eq2 => T1+Wm-WM=0
T1= -g(m-M)
T1= x Newtons

now, plug in T1 in Eq 1,
T2=WM-T1

*Reminder W=mass*9.8

Ans: T2=1.77N

To calculate the tension in the string running over the pulley, we can use the concept of Newton's second law and the principle of conservation of energy. Here are the steps to find the tension:

Step 1: Identify the forces acting on the objects:
- The force of gravity acting on the mass M.
- The force of gravity acting on the mass m.
- The tension in the string connecting the two objects.

Step 2: Set up the equations of motion:
- For the object with mass M:
F_net = M * g - T = M * a1 (equation 1)
- For the object with mass m:
F_net = T - m * g = m * a2 (equation 2)
- Here, g represents the acceleration due to gravity (approximately 9.8 m/s^2), a1 is the acceleration of M, and a2 is the acceleration of m.

Step 3: Express the forces in terms of mass and acceleration:
- Rewrite equation 1 as: T = M * g - M * a1 (equation 3)
- Rewrite equation 2 as: T = m * g + m * a2 (equation 4)

Step 4: Equate the tensions from both equations:
- Setting equations 3 and 4 equal to each other, we have: M * g - M * a1 = m * g + m * a2
- Simplify the equation: M * g - m * g = M * a1 + m * a2
- Rearrange the equation: (M - m) * g = (M * a1) + (m * a2)

Step 5: Apply the conservation of energy:
- Since there is no friction and the system is vertically aligned, the total mechanical energy remains constant.
- The change in gravitational potential energy equals the change in kinetic energy: m * g * h = 0.5 * m * v2^2
- Here, v2 represents the velocity of the mass m.

Step 6: Determine the acceleration:
- For mass M, a1 = 0 since it is connected to the ceiling.
- Solving the conservation of energy equation for v2, we get: v2 = √(2 * g * h)
- Using v = a * t, where t is the time taken to reach the bottom:
- We know the distance traveled is h, so h = 0.5 * a2 * t^2
- Rearrange the equation as: t = √(2h / a2)
- Substitute the value of v2 into the equation: t = √(2h / (2g))

Step 7: Calculate the acceleration a2:
- Substitute the given values into the equation: a2 = 2 * g * h / (2g)
- Simplify the equation: a2 = h

Step 8: Substitute the values into the equation (M - m) * g = (M * a1) + (m * a2):
- T = (m * g) + (m * h)
- T = m * (g + h)
- T = 0.181 kg * (9.8 m/s^2 + h)

Step 9: Convert the mass to kilograms:
- Given that M = 393 g and m = 181 g, convert them to kg: M = 0.393 kg and m = 0.181 kg

Step 10: Calculate the final tension T:
- Plug in the values: T = 0.181 kg * (9.8 m/s^2 + h)

Since the specified height (h) is not given, we cannot provide a specific numerical value for T. You will need to provide the height (h) in order to obtain the tension T in the string running over the pulley.