A chair of weight 85.0N lies atop a horizontal floor; the floor is not frictionless. You push on the chair with a force of F = 45.0N directed at an angle of 34.0∘ below the horizontal and the chair slides along the floor. Using Newton's laws, calculate n, the magnitude of the normal force that the floor exerts on the chair.
To solve this problem using Newton's laws, we need to break down the forces acting on the chair and apply Newton's second law, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration.
The forces acting on the chair are:
1. Weight (W): This is the force due to gravity, given by the product of mass (m) and the acceleration due to gravity (g ≈ 9.8 m/s²). In this case, the weight is given as 85.0N.
2. Applied Force (F): This is the force exerted on the chair by pushing it. Its magnitude is given as 45.0N, and it is directed at an angle of 34.0° below the horizontal.
3. Normal Force (N): This is the force exerted by the floor on the chair perpendicular to the surface. This is the force we need to find.
Since the chair is sliding along the floor, we know that the applied force (F) is greater than the maximum static friction force. The normal force (N) and friction force (f) are related, as the normal force provides the necessary opposing force to prevent the chair from sinking into the floor.
To calculate the normal force (N), we can use the following steps:
1. Resolve the applied force (F) into its horizontal and vertical components.
Fx = F * cos(34.0°) (horizontal component)
Fy = F * sin(34.0°) (vertical component)
2. Calculate the net force in the y-direction:
Net force in the y-direction = Fy - W (upward forces are positive)
= Fy - mg (substituting W = mg)
3. Since the chair is not moving vertically, the net force in the y-direction must equal zero. Thus, the normal force (N) must be equal in magnitude but opposite in direction to the net force:
N = -Net force in the y-direction
Now we can go ahead and calculate the normal force (N).
Given:
Weight (W) = 85.0N
Applied Force (F) = 45.0N
Angle (θ) = 34.0°
Acceleration due to gravity (g) ≈ 9.8 m/s²
Step 1: Resolve the applied force into horizontal and vertical components:
Fx = F * cos(θ)
= 45.0N * cos(34.0°)
≈ 37.071N
Fy = F * sin(θ)
= 45.0N * sin(34.0°)
≈ 24.361N
Step 2: Calculate the net force in the y-direction:
Net force in the y-direction = Fy - W
= 24.361N - 85.0N
= -60.639N
Step 3: Calculate the normal force:
N = -Net force in the y-direction
≈ -(-60.639N)
≈ 60.639N
Therefore, the magnitude of the normal force that the floor exerts on the chair is approximately 60.639N.