The drawing shows a bicycle wheel resting against a small step whose height is h = 0.125 m. The weight and radius of the wheel are W = 23.1 N and r = 0.330 m, respectively. A horizontal force vector F is applied to the axle of the wheel. As the magnitude of vector F increases, there comes a time when the wheel just begins to rise up and loses contact with the ground. What is the magnitude of the force when this happens?
To find the magnitude of the force when the wheel just begins to rise up and loses contact with the ground, we need to consider the equilibrium condition for the wheel.
When the wheel is just about to lift off the ground, the normal force exerted by the ground on the wheel becomes zero. At this point, the weight of the wheel is balanced by the applied force.
To calculate the force required, we can use the following equation:
ΣF = 0
Where ΣF is the sum of the forces in the vertical direction. In this case, the only force acting in the vertical direction is the weight of the wheel, which is W = 23.1 N.
Therefore, we have:
W = 0
Solving for the horizontal force F:
F = W
Substituting the given values:
F = 23.1 N
So, the magnitude of the force required to make the wheel just begin to rise up and lose contact with the ground is 23.1 N.
To find the magnitude of the force when the wheel just begins to rise up and loses contact with the ground, we can analyze the forces acting on the wheel.
When the wheel is in contact with the ground, it experiences a normal force (N) from the ground pushing upward, and the weight (W) of the wheel pushing downward. Since the wheel is at rest, these two forces are equal in magnitude and opposite in direction. So we have N = W.
When the wheel just begins to rise up and loses contact with the ground, the normal force becomes zero. This happens when the weight of the wheel is canceled out by another force.
In this case, the force that can cancel out the weight of the wheel is the component of the applied force (F) perpendicular to the ground. This force component is the horizontal force (F) multiplied by the cosine of the angle between the force vector and the horizontal direction.
So, we can write the equation:
F * cos(θ) = W
To find the magnitude of the force, we can rearrange the equation:
F = W / cos(θ)
Where θ is the angle between the force vector and the horizontal direction.
In this problem, the angle θ is not given. However, from the information provided, we know that the wheel is resting against a small step whose height is h = 0.125 m. We can assume that the force is applied horizontally, parallel to the ground. Therefore, the angle θ is 0 degrees, and the cosine of 0 degrees is 1 (cos(0) = 1).
So we can simplify the equation to:
F = W / cos(0)
F = W
Therefore, the magnitude of the force when the wheel begins to rise up and loses contact with the ground is equal to the weight of the wheel, which is 23.1 N.