how would you go about doing this type of problem thank you
You are trying to raise a bicycle wheel of mass x and radius r up over a curb of height h . To do this, you apply a horizontal force .
1. What is the least magnitude of the force that will succeed in raising the wheel onto
the curb when the force is applied at the center of the wheel?
2. What is the least magnitude of the force that will succeed in raising the wheel onto the curb when the force is applied at the top of the wheel?
3. In which case is less force required?
case (a)
case (b)
To solve this problem step-by-step, we'll follow the given instructions and solve each part individually:
1. To determine the least magnitude of the force required when applied at the center of the wheel, we need to consider the equilibrium conditions and the forces involved. The force applied at the center must be sufficient to overcome both the gravitational force and the force imbalance caused by the curb.
- First, calculate the gravitational force acting on the wheel. Use the formula:
F_grav = mass x g
- Next, determine the force imbalance caused by the curb. This can be calculated using the torque equation:
Torque_curb = Force_curb x radius
- Since the force applied at the center is perpendicular to the curb, it will create no torque. Therefore, the torque caused by the force applied at the center is zero.
- Now, equate the torque caused by the curb to the torque caused by the gravitational force to find the force required:
Torque_grav = Torque_curb
Force_grav x radius = Force_curb x radius
Force_grav = Force_curb
So, the least magnitude of the force required when applied at the center of the wheel is equal to the force imbalance caused by the curb.
2. To find the least magnitude of the force required when applied at the top of the wheel, we can use the same equilibrium conditions and torque equation as before but take into account that the force is now applied at the top.
- The torque caused by the force applied at the top is given by:
Torque_top = Force_top x radius
- Equate the torque caused by the curb to the torque caused by the gravitational force to find the force required:
Torque_grav = Torque_curb
Force_grav x radius = Force_curb x radius + Torque_top
Force_grav = Force_curb + (Torque_top / radius)
So, the least magnitude of the force required when applied at the top of the wheel is equal to the force imbalance caused by the curb plus the torque caused by the force applied at the top.
3. To determine which case (a or b) requires less force, compare the magnitudes of the forces obtained in parts 1 and 2.
- If the magnitude of the force required when applied at the center (from part 1) is greater than the magnitude of the force required when applied at the top (from part 2), then case (b) requires less force.
- If the magnitude of the force required when applied at the center is less than the magnitude of the force required when applied at the top, then case (a) requires less force.
- If the magnitudes are equal, then both cases require the same amount of force.
To solve this type of problem, we need to understand the concept of torque and equilibrium. Torque is the rotational equivalent of force and is calculated by multiplying the force applied to an object by the perpendicular distance from the object's pivot point.
Let's go through the steps for each question:
1. What is the least magnitude of the force that will succeed in raising the wheel onto the curb when the force is applied at the center of the wheel?
To solve this, we need to balance the torque caused by the weight of the wheel (mg) acting at its center with the torque created by the applied force (F). Since the applied force is at the center, it does not create any torque. The torque created by the weight acts around the pivot point, which is the edge of the curb. The torque equation is given by:
Torque = Force x Distance
For the wheel to be balanced, the torque created by the weight and the torque created by the applied force should be equal. Mathematically, we can write:
mg x r = F x h
Rearranging the equation, we can find the least magnitude of the force (F):
F = mg x r / h
2. What is the least magnitude of the force that will succeed in raising the wheel onto the curb when the force is applied at the top of the wheel?
When the force is applied at the top of the wheel, it creates a torque. The torque created by the applied force acts around the pivot point, which is the edge of the curb. In this case, the torque equation becomes:
Torque = Force x Distance
mg x r = F x (h + r)
Again, for the wheel to be balanced, the torque created by the weight and the torque created by the applied force should be equal. Rearranging the equation, we can find the least magnitude of the force (F):
F = (mg x r) / (h + r)
3. In which case is less force required?
To determine which case requires less force, we can compare the previous equations for the least magnitude of the force. By analyzing the equations, we can see that the force required is less when the force is applied at the center of the wheel (F = mg x r / h) compared to when the force is applied at the top of the wheel (F = (mg x r) / (h + r)).
Therefore, the answer to question 3 is that, in case (a), where the force is applied at the center of the wheel, less force is required to raise the wheel onto the curb.
1. Consider the torque (moment) applied to the wheel, as measured about the point where the wheel touches the curb. To make the wheel rise above the curb, the torque due to the force must exceed the torque due to the weight, which equals m g sqrt(R^2-h^2) and is in the opposite direction. There will be no force where the wheel touches the road since the wheel will be rising.
The torque due to the applied force is
F (r-h) when it is applied at the center of the wheel.
I will use m instead of x to represent the mass.
F (r-h) = m g sqrt(r^2-h^2)
sqrt(r^2-h^2) is the horizontal distance from the center of the wheel to the point where the wheel touches the curb. It is the "lever arm" for the torque due to the weight
2. Do this case similarly, but the torque due to the applied force becomes
F(2r-h). This will make the required F less.