Does the Size of the Wheel Affect the Velocity of a Rolling Vehicle?
Have you ever wondered if the size of the wheel affects how fast a vehicle rolls down a hill? Specifically, let’s say you’re sitting on a skateboard, and you’re traveling downhill powered purely by gravity. Would the size of the wheels have any effect on the velocity of the skateboard?
The Start of Our Investigation
There are a lot of variables to consider when answering this question. In this theoretical scenario, let’s assume that the surface of the road is smooth and level and the tires are roughly equivalent in terms of friction and material.
Some have suggested that larger wheels are faster because they offer more springiness, and smaller wheels are slower because they deform more and thus have higher rolling resistance. It’s true that larger wheels have an easier time rolling over imperfections in the road than smaller wheels, which could allow for a smoother ride. But we’re interested in the theoretical aspects of wheel size and velocity, ignoring factors like rolling resistance.
Moment of Inertia and Acceleration
If we look at the theoretically simplest effect of moment of inertia, the acceleration of the vehicle does not depend on the size of the wheel. However, it does depend on the shape of the wheel.
Suppose a wheel has a moment of inertia
I
, which for a solid cylinder would be:
I = 1/2 * m * r^2
Where
m
is the mass of the wheel and
r
is its radius.
The kinetic energy of the wheel is:
K = 1/2 * m * v^2 + 1/2 * I * ω^2
Now suppose you have
N
of these wheels mounted to a chassis of mass
M
. The kinetic energy of the entire ensemble is:
K = 1/2 * (M + Nm) * v^2 + 1/2 * NI * ω^2
If the wheels roll without slipping, then
v = rω
, and we get:
K = 1/2 * (M + Nm + NI/r^2) * v^2
If this whole thing is rolling down a slope of inclination
θ
(where
θ = 0
corresponds to horizontal), the total energy is:
E = 1/2 * (M + Nm + NI/r^2) * v^2 - (M + Nm) * g * d * sin(θ) = 0
Where
g
is the acceleration due to gravity and
d
is the distance traveled along the slope from the point at which the vehicle started rolling. If you take the time derivative of this and do a little algebra, you get:
a = (M + Nm) * g * sin(θ) / (M + Nm + NI/r^2)
It’s worth noting that moment of inertia for any kind of rotationally symmetric object (like a wheel) is given by a formula of the form:
I = bmr^2
Where
b
is some number. For instance,
b = 1/2
for the solid cylinder we looked at earlier. For a hollow wheel (no hubcaps or anything) you’d have
b = 1
, for a sphere
b = 2/5
, and so on. So the acceleration formula can be simplified to:
a = (M + Nm) * g * sin(θ) / (M + Nm(1 + b))
Notice that it doesn’t depend on the size of the wheel (
r
) anymore. However, it does depend on the constant
b
, which is determined by the shape of the wheel. A wheel with a larger value of
b
will undergo less acceleration.
Conclusion
So, in answer to the original question, if a vehicle is rolling down a hill, its speed does not depend on the size of the wheel. However, the shape of the wheel does matter. Wheels with a higher moment of inertia, such as solid cylinders or hollow wheels, will undergo less acceleration than wheels with a lower moment of inertia, such as spheres.
It’s important to keep in mind that this is a simplified theoretical scenario. In real-world situations, factors such as rolling resistance and friction will also affect the velocity of a rolling vehicle. But by looking at the basic principles of moment of inertia and acceleration, we can gain a deeper understanding of the role wheel size and shape play in the movement of vehicles.
If a vehicle is rolling down a hill, will its speed depend on the size of the wheel?
Does the Size of the Wheel Affect the Velocity of a Rolling Vehicle?
Have you ever wondered if the size of the wheel affects how fast a vehicle rolls down a hill? Specifically, let’s say you’re sitting on a skateboard, and you’re traveling downhill powered purely by gravity. Would the size of the wheels have any effect on the velocity of the skateboard?
The Start of Our Investigation
There are a lot of variables to consider when answering this question. In this theoretical scenario, let’s assume that the surface of the road is smooth and level and the tires are roughly equivalent in terms of friction and material.
Some have suggested that larger wheels are faster because they offer more springiness, and smaller wheels are slower because they deform more and thus have higher rolling resistance. It’s true that larger wheels have an easier time rolling over imperfections in the road than smaller wheels, which could allow for a smoother ride. But we’re interested in the theoretical aspects of wheel size and velocity, ignoring factors like rolling resistance.
Moment of Inertia and Acceleration
If we look at the theoretically simplest effect of moment of inertia, the acceleration of the vehicle does not depend on the size of the wheel. However, it does depend on the shape of the wheel.
Suppose a wheel has a moment of inertia
, which for a solid cylinder would be:
Where
is the mass of the wheel and
is its radius.
The kinetic energy of the wheel is:
Now suppose you have
of these wheels mounted to a chassis of mass
. The kinetic energy of the entire ensemble is:
If the wheels roll without slipping, then
, and we get:
If this whole thing is rolling down a slope of inclination
(where
corresponds to horizontal), the total energy is:
Where
is the acceleration due to gravity and
is the distance traveled along the slope from the point at which the vehicle started rolling. If you take the time derivative of this and do a little algebra, you get:
It’s worth noting that moment of inertia for any kind of rotationally symmetric object (like a wheel) is given by a formula of the form:
Where
is some number. For instance,
for the solid cylinder we looked at earlier. For a hollow wheel (no hubcaps or anything) you’d have
, for a sphere
, and so on. So the acceleration formula can be simplified to:
Notice that it doesn’t depend on the size of the wheel (
) anymore. However, it does depend on the constant
, which is determined by the shape of the wheel. A wheel with a larger value of
will undergo less acceleration.
Conclusion
So, in answer to the original question, if a vehicle is rolling down a hill, its speed does not depend on the size of the wheel. However, the shape of the wheel does matter. Wheels with a higher moment of inertia, such as solid cylinders or hollow wheels, will undergo less acceleration than wheels with a lower moment of inertia, such as spheres.
It’s important to keep in mind that this is a simplified theoretical scenario. In real-world situations, factors such as rolling resistance and friction will also affect the velocity of a rolling vehicle. But by looking at the basic principles of moment of inertia and acceleration, we can gain a deeper understanding of the role wheel size and shape play in the movement of vehicles.