two blocks of each mass m=3.75kg are fastened to the ceiling of an elevator. If the elevator accelerates upward at 1.6m/s^2.
Find the tension in T1 and T2 in each upper and lower rope
Use F=ma
For top string,
m=2*3.75=7.5 kg
a=1.6 m/s62
T1=force in top string
For staionary masses, T=mg
If in addition, T1 casues upward acceleration, then
(T1-mg)=ma
or T1=m(a+g)=7.5(1.6+9.81)=85.75 N
For the lower string (T2), use same argument, but with a different mass.
To find the tension in T1 and T2, we need to consider the forces acting on each block in the elevator.
Let's start with Block 1, which is the upper block.
The forces acting on Block 1 are:
1. The force of gravity (mg) pulling it downward.
2. The tension force in T1 pulling it upward.
3. The force of acceleration, which in this case is the elevator's acceleration (ma) also pulling it downward.
Using Newton's second law (F = ma), we can write down the equation of motion for Block 1:
T1 - mg = ma
Now, let's move on to Block 2, which is the lower block.
The forces acting on Block 2 are:
1. The force of gravity (mg) pulling it downward.
2. The tension force in T2 pulling it upward.
3. The force of acceleration, which in this case is the elevator's acceleration (ma) also pulling it downward.
Again, using Newton's second law, we can write down the equation of motion for Block 2:
T2 - mg = ma
Since the masses of both blocks are the same (m = 3.75 kg), the value of m and g will cancel out when we subtract the equations of Block 2 from Block 1 to eliminate the acceleration term.
(T1 - mg) - (T2 - mg) = ma - ma
T1 - T2 = 0
Therefore, T1 = T2.
Knowing that T1 = T2, we can solve for the tension in T1 or T2.
Let's substitute T1 = T2 in any one of the equations of motion, let's say the equation for Block 1:
T1 - mg = ma
Substituting T1 = T2, we get:
T2 - mg = ma
To find the tension in T1 or T2, we need to calculate the value of mg and ma.
For mg, we use the formula: mg = mass * acceleration due to gravity (g).
Given that mass (m) is 3.75 kg, and acceleration due to gravity (g) is approximately 9.8 m/s^2, we can calculate mg:
mg = 3.75 kg * 9.8 m/s^2 = 36.75 N
For ma, we use the formula: ma = mass * acceleration (a).
Given that mass (m) is 3.75 kg, and acceleration (a) is 1.6 m/s^2 (the elevator's acceleration in this case), we can calculate ma:
ma = 3.75 kg * 1.6 m/s^2 = 6 N
Now we can calculate the tension in T1 or T2 using the equation:
T2 - mg = ma
Substituting the values, we get:
T2 - 36.75 N = 6 N
Now, solve for T2:
T2 = 6 N + 36.75 N
T2 = 42.75 N
Since T1 = T2, the tension in T1 is also 42.75 N.
Therefore, the tension in T1 and T2 (both upper and lower ropes) is 42.75 N.