The figure shows a bicycle wheel resting against a small step whose height is h = 0.120 m. The weight and radius of the wheel are W = 20.0 N and r = 0.340 m. A horizontal force is applied to the axle of the wheel. As the magnitude of 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 determine the point at which the normal force becomes zero. At this point, the weight of the wheel is balanced by the applied force.
1. Start by drawing a free-body diagram for the wheel. The weight of the wheel acts downward, and the normal force acts upward.
- The weight of the wheel (W) = 20.0 N.
- The normal force (N) acts perpendicular to the surface of the step.
2. Since the wheel is in equilibrium, the sum of the forces in the vertical direction must be zero. This gives us the equation: N - W = 0.
3. The normal force (N) can be expressed in terms of the applied force (F) and the weight of the wheel (W) using trigonometry.
- The angle between the normal force and the weight is the angle between the wheel and the horizontal surface, which is given by the arctan of the height of the step (h) divided by the radius of the wheel (r). So, θ = arctan(h/r).
- The normal force (N) can be written as N = W * cos(θ).
4. Substitute the value of N from step 3 into the equation from step 2: W * cos(θ) - W = 0.
5. Rearrange the equation to solve for the applied force (F): F = W / cos(θ).
6. Substitute the given values into the equation: F = 20.0 N / cos(arctan(0.120 m / 0.340 m)).
7. Use a calculator to evaluate the expression: F ≈ 20.0 N / cos(19.64°).
8. The magnitude of the force when the wheel just begins to rise up is approximately equal to the calculated value of F.
Therefore, the magnitude of the force when the wheel just begins to rise up and loses contact with the ground is approximately equal to the calculated value of F, which can be found by evaluating the expression F ≈ 20.0 N / cos(arctan(0.120 m / 0.340 m)).
To find the magnitude of the force when the wheel loses contact with the ground, we can use the concept of equilibrium. When the wheel just begins to rise up, the gravitational force pulling the wheel downwards is balanced by the normal force exerted by the ground upwards, and also balanced by the force applied to the axle horizontally.
The weight of the wheel acts downwards and can be calculated using the formula:
Weight (W) = mass (m) × gravitational acceleration (g)
In this case, the weight of the wheel is given as 20.0 N.
Now, let's consider the forces acting on the wheel when it is in contact with the ground. There are three forces: the weight acting downwards, the normal force acting upwards, and the applied force acting horizontally.
When the wheel just begins to rise up, the normal force becomes zero since there is no longer any contact with the ground. Therefore, the gravitational force is equal in magnitude to the applied force:
Weight (W) = Applied force (F)
Now, we can substitute the weight of the wheel into the equation:
20.0 N = Applied force (F)
Hence, the magnitude of the force when the wheel just begins to rise up and loses contact with the ground is 20.0 N.