A ninety kilogram sofa is being pushed by four meters to the other side of a hall. After an initial push the sofa is pushed with a 600 N force. In this case, 0.6 is the coefficient of kinetic friction. (The diagram shows the sofa being pushed from the right).

1) What is the value and direction of the total force on the sofa?

I got 70.26 Newtons but don't know how to get the direction.

2) What is the value and direction of the acceleration of the sofa?

.78m/s^2, but don't know the direction again.

3. How long would it take to push the sofa 4 meters across?

Not sure what I am supposed to use to solve this.

total force on the sofa:

a. floor holding it up: 90*9.8N
b. person pushing on it: 600N
c. Fricion from floor: .6*90

ok, net horizontal force is 600-54
net vertical force= 882N
so net force=sqrt(882^2+546^2)
direction: arc tan (882/546)above the horizontal.

acceleration: 546/90 horizontal.

To answer these questions, we need to use Newton's second law of motion, which states that the net force on an object is equal to the product of its mass and acceleration (F = ma). We also need to consider the forces acting on the sofa.

1) To find the value and direction of the total force on the sofa, we need to consider the forces acting on it. The force applied to the sofa is 600 N, but there is also a force opposing its motion due to friction. The frictional force can be calculated using the equation: frictional force = coefficient of friction * normal force. The normal force is equal to the weight of the sofa, which is the mass multiplied by the acceleration due to gravity (9.8 m/s^2).

Normal force = mass * acceleration due to gravity
Normal force = 90 kg * 9.8 m/s^2
Normal force = 882 N

Frictional force = coefficient of friction * normal force
Frictional force = 0.6 * 882 N
Frictional force = 529.2 N

The total force on the sofa is the applied force minus the frictional force.

Total force = applied force - frictional force
Total force = 600 N - 529.2 N
Total force = 70.8 N

To determine the direction of the total force, we need to know the direction of the applied force. If the applied force is in the opposite direction to the motion of the sofa (as indicated by the diagram), then the total force will also be in the opposite direction, i.e., to the right.

2) The acceleration of the sofa can be calculated using Newton's second law of motion. Rearranging the equation, we get: acceleration = net force / mass.

Acceleration = Total force / mass
Acceleration = 70.8 N / 90 kg
Acceleration ≈ 0.7878 m/s^2

The direction of the acceleration will be in the same direction as the net force, which in this case is to the right (since the applied force is in the opposite direction to the motion).

3) To determine how long it would take to push the sofa 4 meters across, we need to use the equations of motion. The relevant equation, in this case, is: distance = initial velocity * time + 0.5 * acceleration * time^2. However, since we don't know the initial velocity and assume it is at rest, the equation simplifies to: distance = 0.5 * acceleration * time^2.

Rearranging the equation, we can solve for time:

time^2 = (2 * distance) / acceleration
time = √ [(2 * 4 m) / 0.7878 m/s^2]
time ≈ 3.2 seconds.

Therefore, it would take approximately 3.2 seconds to push the sofa 4 meters across.