A 5kg object placed on a frictionless, horizontal table connected to a string that passes over a pulley and then is fastened to a hanging 9kg object. Draw a free body diagram of both the objects. Find the acceleration of the two objects and the tension in the strings.
50 m\ s
To draw a free body diagram (FBD) for each object, we need to consider the forces acting on them.
1. For the 5kg object on the table:
- There is the force of gravity acting downwards (mg).
- Since the table is frictionless, there is no frictional force.
- There is tension in the string pulling the object towards the pulley.
2. For the 9kg hanging object:
- There is the force of gravity acting downwards (mg).
- There is tension in the string pulling the object upwards.
To find the acceleration and tension in the strings, we can use Newton's second law and consider the system as a whole.
Let's assume:
- The acceleration of the system is denoted as 'a'.
- The tension in the string connected to the 5kg object is denoted as 'T1'.
- The tension in the string connected to the 9kg object is denoted as 'T2'.
Now, applying Newton's second law to the 5kg object:
ΣF = ma
1. For the 5kg object:
- T1 - mg = ma
Next, apply Newton's second law to the 9kg object:
ΣF = ma
2. For the 9kg object:
- T2 - mg = 9a
Now, consider the system as a whole:
The tension in the string is the same on both sides (T1 = T2). This is because the string is assumed to be massless and has no internal friction.
Combining the two equations:
T1 - 5g = 5a (equation 1)
T2 - 9g = 9a (equation 2)
Since T1 = T2, we can rewrite equation 1 as:
T2 - 5g = 5a (equation 3)
Now, let's solve the system of equations (2 and 3) to find the acceleration and tension.
Subtract equation 3 from equation 2:
T2 - 9g - T2 + 5g = 9a - 5a
4g = 4a
a = g
The acceleration of the system is equal to the acceleration due to gravity, which is approximately 9.8 m/s^2.
To find the tension in the strings, we can substitute the value of 'a' in either equation 1 or 2.
Using equation 1:
T1 - 5g = 5a
T1 - 5g = 5 * 9.8
T1 = 49 N
Therefore, the tension in the string connected to the 5kg object (T1) is 49 N.
Since T1 = T2, the tension in the string connected to the 9kg object (T2) is also 49 N.