A box of mass m2 = 3.1 kg rests on a frictionless horizontal shelf and is attached by strings to boxes of masses m1 = 1.6 kg and m3 = 2.6 kg as shown below. Both pulleys are frictionless and massless. The system is released from rest. After it is released, find the tension in each string
m1•a=T1-m1•g,
m2•a=T3-T1,
m3•a=m3•g – T3
(m1+m2+m3) •a= T1-m1•g+ T3-T1+m3•g – T3= g• (m3-m1)
a= g• (m3-m1)/ (m1+m2+m3) =9.8•(2.6-1.6)/(1.6+3.1+2.6)=1.34 m/s²
T1=m1• (a+g) = 1.6•(1.34+9.8)=17.8 N
T3 = m3• (g-a) = 2.6(9.8-1.34) = 22 N
To find the tension in each string, we need to consider the forces acting on each box.
Let's consider box 1 first. It is connected to box 2 by a string, and there is tension in this string. We'll call this tension T1.
Since the pulley is frictionless and massless, the tension in the string is the same on both sides of the pulley. So T1 is also the tension in the string connected to box 3.
Now, let's consider box 2. It is connected to box 1 and box 3 by strings. The force acting on box 2 is its weight (mg2) downward, and it is balanced by the tension in the string connected to box 1 (T1) and the tension in the string connected to box 3 (T2). So we have:
mg2 = T1 + T2
Similarly, for box 3, the force acting on it is its weight (mg3) downward, and it is balanced by the tension in the string connected to box 2 (T2) and the tension in the string connected to the shelf (T3). So we have:
mg3 = T2 + T3
Now, let's write down the equations involving the given masses:
m1 = 1.6 kg
m2 = 3.1 kg
m3 = 2.6 kg
We can solve these two equations together to find the values of T1 and T2. Here's how:
1) From the equation mg2 = T1 + T2, we can substitute mg2 with m2g since g is the acceleration due to gravity:
m2g = T1 + T2
2) From the equation mg3 = T2 + T3, we can substitute mg3 with m3g:
m3g = T2 + T3
3) Now we'll substitute the values of m2, m3, and g:
(3.1kg)(9.8m/s^2) = T1 + T2
(2.6kg)(9.8m/s^2) = T2 + T3
4) Now we can solve the equations simultaneously to find T1, T2, and T3.
Solving these equations will give you the values of T1, T2, and T3, which represent the tension in each string.
Note: The system is released from rest, which means it starts from a state of no initial velocity. Therefore, we are assuming that the masses are not accelerating at the beginning.