You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this you apply a horizontal force F. What is the smallest magnitude of the force F that will succeed in raising the wheel onto the curb when the force is applied
(a) at the center of the wheel and (b) at the top of the wheel? (c) in which case is less force required?
Find the answer in your previous posts.
To determine the smallest magnitude of the force required to raise the bicycle wheel onto the curb, we can analyze the forces acting on the wheel and use Newton's laws of motion.
Let's start with case (a) where the force is applied at the center of the wheel.
In this case, the force F acts at the center of mass of the wheel. To raise the wheel, the force must be large enough to overcome the gravitational force pulling the wheel downward and to generate enough torque to rotate the wheel over the curb.
1. Gravitational force: The gravitational force pulling the wheel downward can be calculated using the equation F_gravity = m * g, where m is the mass of the wheel and g is the acceleration due to gravity.
2. Torque: The torque exerted by the applied force can be calculated as torque = F * R, where R is the radius of the wheel.
The minimum magnitude of the force required to raise the wheel is when it just balances the gravitational force and provides enough torque to rotate the wheel over the curb. Therefore, the magnitude of the force F can be calculated as:
F = (m * g) / R
Now let's consider case (b) where the force is applied at the top of the wheel.
In this case, the force is applied at the highest point of the wheel. The force must be large enough to overcome not only the gravitational force but also the additional torque due to the offset location.
1. Gravitational force: The gravitational force pulling the wheel downward remains the same as in case (a), F_gravity = m * g.
2. Torque: The torque due to the applied force is torque = F * (2R), as the force is applied at a distance of 2R from the axle.
Again, the minimum magnitude of the force required to raise the wheel is when it just balances the gravitational force and provides enough torque to rotate the wheel over the curb. Therefore, the magnitude of the force F can be calculated as:
F = (m * g) / (2R)
Now, to answer part (c) of the question, we need to compare the magnitudes of the forces from cases (a) and (b).
Comparing the two force equations, we can observe that the force required in case (b) is twice the force required in case (a):
F(b) = F(a) / 2
Therefore, less force is required when the force is applied at the center of the wheel (case a) rather than at the top of the wheel (case b).
To summarize:
(a) The minimum magnitude of the force required when applied at the center of the wheel is F(a) = m * g / R.
(b) The minimum magnitude of the force required when applied at the top of the wheel is F(b) = m * g / (2R).
(c) Less force is required when the force is applied at the center of the wheel (case a) compared to the force applied at the top of the wheel (case b).
To determine the smallest magnitude of the force F required to raise the bicycle wheel onto the curb, we can analyze the forces acting on the wheel.
(a) When the force is applied at the center of the wheel:
In this case, the force F acts perpendicular to the radius of the wheel. To lift the wheel onto the curb, this force must overcome the gravitational force acting on the wheel.
The force required to lift the wheel at the center can be calculated using the following equation:
F = m * g,
where m is the mass of the wheel and g is the acceleration due to gravity.
(b) When the force is applied at the top of the wheel:
In this case, the force F acts tangentially to the wheel. When the force is applied at the top, there will be a downward component of the force due to the angle at which it acts. This downward component must be overcome by the upward force exerted by the curb to lift the wheel.
The minimum force required to lift the wheel at the top can be calculated using the following equation:
F = (m * g) / cos(theta),
where theta is the angle between the force and the vertical direction.
(c) Comparing the two cases:
When the force is applied at the center of the wheel, the force required is simply the weight of the wheel, F = m * g. This is the minimum force required.
When the force is applied at the top of the wheel, there is an additional component of the force acting downwards due to the angle. Therefore, a greater magnitude of force is required compared to the center.
In conclusion, less force is required when the force is applied at the center of the wheel compared to when it is applied at the top.