Consider two solid uniform cylinders that have the same mass and length, but different radii: the radius of cylinder A is much smaller than the radius of cylinder B. Rolling down the same incline, which one of the two cylinders will reach the bottom first? Assume both cylinders are rolling without slipping (pure roll).
a) cylinder A
b)cylinder B
c)both in same time.
I=mr²/2
mgh= mv²/2 +Iω²/2 =
=mv²/2 +mr²v²/r²4 =
= mv²/2+ mv²/4
=3mv²/4
v=sqrt{4gh/3} =>
c)both in same time.
To determine which cylinder will reach the bottom of the incline first, we need to consider their moments of inertia.
The moment of inertia depends on the mass distribution and shape of an object. For a solid cylinder, the moment of inertia is given by the formula:
I = (1/2) * m * r^2
where I is the moment of inertia, m is the mass, and r is the radius of the cylinder.
In this case, we have two cylinders with the same mass and length, but different radii. Since both cylinders are rolling without slipping (pure roll), their moments of inertia are directly proportional to their radii squared.
Let's assume that the mass of cylinder A is m and its radius is rA, while the mass of cylinder B is also m but with a larger radius rB (rB > rA).
Since rB > rA, the moment of inertia of cylinder B will be larger than that of cylinder A. Therefore, cylinder B will have a larger resistance to rotational motion compared to cylinder A.
When rolling down an incline, the cylinders will start with the same potential energy. As they roll down, this potential energy is converted into kinetic energy, both translational and rotational.
The total kinetic energy can be expressed as:
K = (1/2) * mv^2 + (1/2) * I * ω^2
where K is the kinetic energy, m is the mass, v is the translational velocity, I is the moment of inertia, and ω is the angular velocity.
Since both cylinders are rolling without slipping, their translational velocity and angular velocity are related by the equation v = ω * r (where r is the radius of the cylinder).
Since the cylinders have the same mass, they will have the same translational velocity as they roll down the incline. However, the cylinder with the smaller moment of inertia (cylinder A) will have a larger angular velocity compared to the cylinder with the larger moment of inertia (cylinder B) for the same translational velocity.
This means that cylinder A will have a higher rotational speed compared to cylinder B. As a result, cylinder A will reach the bottom of the incline first.
Therefore, the answer is:
a) Cylinder A.