Three objects are connected by two very light, flexible, and unstretchable string as shown in the figure, where M=9 kg and m=2.5 kg. You can ignore friction and the mass of the pulley. When the objects are released;
(a) Calculate the magnitude of the acceleration (in m/s2) of each object.
(b) Calculate the tension T1 (in N) in the string.
(c) Calculate the tension T2 (in N) in the string.
To calculate the magnitude of the acceleration for each object, 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. Let's consider each object separately.
Object M:
The forces acting on object M are its weight (mg) pulling downward and the tension force T1 pulling upward. Since there is no friction and the pulley is massless, these are the only forces acting on object M. Therefore, the net force is given by:
Net force on M = T1 - mg
Using Newton's second law, we can write the equation as:
m * aM = T1 - mg
For Object m:
The forces acting on object m are its weight (mg) pulling downward and the tension force T2 pulling upward. Again, since there is no friction and the pulley is massless, these are the only forces acting on object m. Therefore, the net force is given by:
Net force on m = T2 - mg
Using Newton's second law, we can write the equation as:
m * am = T2 - mg
Now, let's solve the equations:
For object M: 9 * aM = T1 - 9 * 9.8
For object m: 2.5 * am = T2 - 2.5 * 9.8
To calculate the tension T1 in the string, we need to find the value of aM. Let's solve the equations simultaneously:
9 * aM = T1 - 88.2 (equation 1)
2.5 * am = T2 - 24.5 (equation 2)
Since the two strings are unstretchable, the acceleration of object m is equal to the acceleration of object M. Therefore, we can write: am = aM
Now, let's substitute am = aM into equation 2:
2.5 * aM = T2 - 24.5 (equation 3)
Combining equation 1 and equation 3, we have:
9 * aM = T1 - 88.2
2.5 * aM = T2 - 24.5
Since we know the value of masses (M = 9 kg, m = 2.5 kg) and acceleration is the same for both objects, we can solve these equations to find aM.
Now, let's solve these simultaneous equations using algebraic methods or substitution methods to find the value of aM.
Once we have the value of aM, we can calculate the tensions T1 and T2 using the equations:
T1 = 9 * aM + 88.2
T2 = 2.5 * aM + 24.5
I hope this helps! Let me know if you need further assistance.