In the diagram, a mass of 12kg hanging from a light inextensible string is pulled aside by a horizontal force, R, such that the string is inclined at 45\(^o\) to the vertical. If the system is in equilibrium, calculate the;
To solve this problem, we need to use the principles of equilibrium. This means that the forces acting on the mass must balance out, so that the net force is zero and the mass remains stationary.
First, let's draw a free-body diagram of the mass. This diagram shows all the forces acting on the mass and their directions. We have two forces acting on the mass: the weight of the mass, which acts downwards, and the tension in the string, which acts upwards and at an angle of 45 degrees to the vertical.
Next, we can resolve the forces into their vertical and horizontal components. The weight of the mass has a vertical component of 12g (where g is the acceleration due to gravity, approximately 9.81 m/s²) acting downwards, and no horizontal component. The tension in the string has both vertical and horizontal components. The vertical component is T cos(45), where T is the tension in the string, and the horizontal component is T sin(45).
Since the mass is in equilibrium, the net force in the horizontal direction must be zero. This means that the horizontal component of the tension in the string must be equal and opposite to the horizontal force R:
T sin(45) = R
Since we know the value of the mass and the acceleration due to gravity, we can calculate the weight of the mass and the vertical component of the tension in the string. The net force in the vertical direction must also be zero, so the vertical component of the tension in the string must balance out the weight of the mass:
T cos(45) = 12g
We can now solve these two equations simultaneously to find the tension in the string and the value of R.
To do this, we can first use the second equation to find the value of T:
T = 12g / cos(45)
T = 12g / 0.707
T = 169.7 N (to two significant figures)
We can then substitute this value of T into the first equation to find the value of R:
R = T sin(45)
R = 169.7 N x 0.707
R = 120 N (to two significant figures)
Therefore, the tension in the string is 169.7 N and the value of R is 120 N.
In summary, we can solve this problem by drawing a free-body diagram, resolving the forces into their components, and using the principles of equilibrium to find the values of the tension in the string and the horizontal force R.
