A person pushes with force F on a box which remains at rest on a rough floor as shown in the top figure. The bottom figures illustrates all the force acting on the box.
a) What condition must be satisfied by all the force acting on the box so that it remains at rest? Draw a vector diagram that illustrates how the forces must combine to satisfy this condition.
b) Suppose the box has weight W = 100 N and the friction force magnitude f is 80% of the normal force magnitude N. That is, f = 0.8N. If the person pushes down at an angle of θ = 35◦ determine the magnitude of the force F the person exerts on the box.
a) In order for the box to remain at rest, the net force acting on it must be zero. This means that the vector sum of all the forces acting on the box must be zero. Mathematically, this can be written as:
∑F = 0
Where ∑F represents the vector sum of all the forces.
To illustrate this condition, we can draw a vector diagram. Since the box is at rest, the force of gravity (weight) must be balanced by the normal force and the force of friction. The normal force N acts perpendicular to the floor, the force of gravity W acts vertically downward, and the frictional force f acts opposite to the direction of the applied force F.
N
↑
|
F → → |
□ -------→ f
|
↓
W
The vector sum of N and f should be equal in magnitude and opposite in direction to W. The vector sum of F, N, f, and W should be zero.
b) Given:
Weight of the box (W) = 100 N
Friction force (f) = 0.8N
Angle of applied force (θ) = 35°
To determine the magnitude of the force F the person exerts on the box, we need to consider the vertical and horizontal components of the force F.
Vertical Component:
The vertical component of force F can be found using the equation:
F_vertical = F * sin(θ)
Horizontal Component:
The horizontal component of force F can be found using the equation:
F_horizontal = F * cos(θ)
Since the box is at rest, the vertical component of force F must balance the weight W:
F_vertical = W
Substituting the given values:
F * sin(θ) = 100
Solving for F, we have:
F = 100 / sin(θ)
To find the magnitude of the force F, we need to substitute the value of θ:
F = 100 / sin(35°)
Using a calculator, we can determine the value of F to be approximately 182.43 N.
a) In order for the box to remain at rest, the sum of all the forces acting on it should be zero. This condition is known as equilibrium. Mathematically, this can be expressed as:
ΣF = 0
Where ΣF represents the vector sum of all the forces.
To illustrate how the forces must combine to satisfy this condition, we can draw a vector diagram. First, draw the weight vector (W) pointing vertically downwards. Then, draw the normal force vector (N) pointing vertically upwards, as it is equal in magnitude and opposite in direction to the weight (according to Newton's third law).
Next, draw the friction force vector (f), which opposes the motion and acts parallel to the contact surface. Its direction will depend on the specific setup of the problem.
Finally, draw the force vector exerted by the person (F). Its magnitude and direction will also depend on the problem statement.
The vector diagram should show that the vector sum of all the forces (ΣF) is zero, indicating equilibrium.
b) Given that the box has a weight (W) of 100 N and the friction force (f) is 80% of the normal force (N), we can determine the magnitudes of these forces.
Since the weight (W) acts vertically downwards, its magnitude is 100 N.
The friction force (f) is given as 80% of the normal force (N). Mathematically, we can express this as:
f = 0.8N
We can rewrite this equation in terms of the weight (W) by using the fact that the weight is equal to the normal force (N):
N = W
Substituting this into the equation for the friction force, we get:
f = 0.8W
Substituting the given weight value of 100 N, we find:
f = 0.8 * 100
f = 80 N
Now, to determine the magnitude of the force (F) exerted by the person, we need to consider the force components in the horizontal and vertical directions.
The vertical component of the force (F) is given by:
F_vertical = F * sin(θ)
The horizontal component of the force (F) is given by:
F_horizontal = F * cos(θ)
Since the box remains at rest, the vertical components of the forces (F_vertical and W) must balance each other:
F_vertical + W = 0
Substituting the weight value of 100 N, we have:
F * sin(θ) + 100 = 0
Solving for F, we get:
F = -100 / sin(θ)
Given that the angle (θ) is 35 degrees, we can substitute this value to find the magnitude of the force (F):
F = -100 / sin(35)
Using a scientific calculator, we can calculate the value of F to be approximately -168.67 N.
Note that the negative sign indicates that the force (F) is acting in the opposite direction to the weight (W).
Therefore, the magnitude of the force (F) exerted by the person is approximately 168.67 N.