Two movers are attempting to push a couch up a ramp. One movers applies a force of 80 N to the right, but the force of friction between the couch and the ramp is providing a force of 110 N in the opposite direction, and the couch slides down the ramp. At minimum, how much force will the second mover need to apply to help push the couch up the ramp?

Answer 1

To determine the minimum force required by the second mover to push the couch up the ramp, we need to analyze the forces acting on the couch.

1. The first mover applies a force of \( F_1 = 80 , \text{N} \) to the right.
2. The frictional force \( F_f = 110 , \text{N} \) opposes the motion and acts to the left.

The net force acting on the couch can be calculated by adding these forces together, taking into account their directions:

\[ \text{Net Force} = F_1 - F_f \]

Inserting the values:

\[ \text{Net Force} = 80 , \text{N} - 110 , \text{N} = -30 , \text{N} \]

A negative net force indicates that the couch is sliding down the ramp with an additional force of \( 30 , \text{N} \) downward (in the direction opposite to the applied forces).

To move the couch upwards, the combined forces exerted by both movers must exceed the frictional force. Let's denote the force exerted by the second mover as \( F_2 \).

For the couch to move upwards, the following inequality must be satisfied:

\[ F_1 + F_2 > F_f \]

Replacing \( F_1 \) and \( F_f \) with the given values:

\[ 80 , \text{N} + F_2 > 110 , \text{N} \]

Now, we can solve for \( F_2 \):

\[ F_2 > 110 , \text{N} - 80 , \text{N} \] \[ F_2 > 30 , \text{N} \]

Hence, the minimum force that the second mover needs to apply in order to help push the couch up the ramp is:

\[ \boxed{30 , \text{N}} \]