Consider a block of mass 2 kg connected to a freely hanging block of mass 4 kg by a light and inextensible string which passes over a light pulley at the edge of a table. The 2 kg mass is on the surface of the table which is assumed to be smooth. Calculate the acceleration of the system and the tension in the string
To solve this problem, we can apply Newton's second law of motion to both blocks individually.
For the 2 kg block:
The only force acting on the 2 kg block is the tension force from the string. According to Newton's second law, we have:
\(F = ma\)
where F is the net force on the block, m is the mass of the block, and a is the acceleration of the block.
The tension force T is pulling the block to the right, so we have:
\(F = T\)
Plugging this into the equation gives us:
\(T = ma\) ----(1)
For the 4 kg block:
The only force acting on the 4 kg block is the gravitational force. The gravitational force equals the mass of the block times the acceleration due to gravity (9.8 m/s²):
\(F = mg\)
where F is the net force on the block, m is the mass of the block, and g is the acceleration due to gravity.
The tension force T is also acting on the 4 kg block, but in the opposite direction to the gravitational force. This gives us:
\(F = mg - T\)
Plugging this into the equation with the tension force from equation (1), we get:
\(mg - T = ma\)
Substituting the values of 4 kg for m, 9.8 m/s² for g, and T from equation (1), gives us:
\(4(9.8) - T = 4a\) ----(2)
Now we have a system of two equations (equations (1) and (2)) with two unknowns (a and T). We can solve these equations simultaneously.
From equation (1):
\(T = 2a\) ----(3)
Substituting equation (3) into equation (2), we have:
\(4(9.8) - 2a = 4a\)
Simplifying and solving for a, we get:
\(39.2 = 6a\)
\(a = 39.2/6\)
\(a = 6.533\) m/s²
Therefore, the acceleration of the system is \(6.533\) m/s².
Substituting the value of a into equation (3), we can find the tension in the string:
\(T = 2(6.533)\)
\(T = 13.066\) N
Therefore, the tension in the string is \(13.066\) N.