A mover has to move a heavy sofa of mass 66 kg to the second floor of the house. As he pulls on the rope tied to the sofa he makes sure that the rope is parallel to the surface of the ramp, which is at 30.0° to the horizontal. If the coefficient of kinetic friction between the sofa and the ramp is 0.400, and the sofa has an acceleration of 0.700 m/s2, find the tension in the rope.
N
-330.39
To find the tension in the rope, we need to consider the forces acting on the sofa. The forces in this case are the gravitational force (mg), the normal force (N), and the frictional force (fk).
1. Start by finding the gravitational force acting on the sofa:
Gravitational force (mg) = mass (m) * acceleration due to gravity (g)
where mass (m) = 66 kg and acceleration due to gravity (g) = 9.8 m/s²
Therefore, mg = 66 kg * 9.8 m/s² = 646.8 N
2. We can break the gravitational force into two components: one parallel to the ramp (mg*sinθ) and one perpendicular to the ramp (mg*cosθ), where θ is the angle of the ramp (30° in this case).
3. Determine the normal force (N) acting perpendicular to the ramp. Since the sofa is on the ramp, the normal force is equal in magnitude and opposite in direction to the perpendicular component of the gravitational force:
N = mg*cosθ = 646.8 N * cos(30°) = 646.8 N * 0.866 = 560.6 N
4. Calculate the frictional force (fk) using the coefficient of kinetic friction (μk) and the normal force (N):
fk = μk * N = 0.400 * 560.6 N = 224.24 N
5. The tension in the rope is equal to the parallel component of the gravitational force minus the frictional force:
Tension (T) = mg * sinθ - fk
T = 646.8 N * sin(30°) - 224.24 N
T = 323.4 N - 224.24 N
T = 99.16 N
Therefore, the tension in the rope is 99.16 N.