Two masses are connected by a string as shown in the figure. Mass = 3.4 rests on a frictionless inclined plane, while = 5.5 is initially held at a height of = 0.90 above the floor.
To solve this problem, we need to find the tension in the string. We can start by analyzing the forces acting on the masses.
1. Mass m1 (3.4 kg) on the inclined plane:
Considering the motion along the inclined plane, we have the following forces:
- The gravitational force, mg, acting vertically downwards.
- The normal force, N, acting perpendicular to the plane.
The component of the gravitational force along the inclined plane is mg sin(θ), where θ is the angle of the inclined plane. Since the inclined plane is frictionless, we don't have to consider the force of friction.
2. Mass m2 (5.5 kg) hanging in the air:
Considering the motion in the vertical direction, we have the following forces:
- The gravitational force, mg, acting vertically downwards.
- The tension force, T, acting vertically upwards.
Since both masses are connected by a string, they experience the same tension force. Therefore, we can equate the tensions in the string for both masses:
T = T
Now, let's find the value of T in terms of m1, m2, and other given quantities:
For mass m1 (on the inclined plane):
The force along the inclined plane is given by F = m1 * g * sin(θ).
For mass m2 (hanging in the air):
The force in the upward direction is given by F = m2 * g.
Setting these two forces equal to each other:
m1 * g * sin(θ) = m2 * g
Now, we can cancel out the gravitational force and solve for θ:
sin(θ) = m2 / m1
Taking the inverse sine to get θ:
θ = sin^(-1)(m2 / m1)
Next, we need to find the height h that mass m2 (5.5 kg) falls. We are given the initial height h0 (0.90 m) above the floor. When mass m2 falls, the change in height is given by:
Δh = h0 - h
Finally, to find the tension force T in the string, we can use the equation:
T = m2 * g + m2 * Δh
Substituting the given values, solve for T.