The objects listed below are placed at the top of a ramp, and roll down to the bottom without slipping. Assuming that there is no air resistance, rank them in order from fastest average rolling speed to slowest.
Penny, large marble, basketball, and hula hoop
If they do not slip, there is a friction force up the slope resulting in angular acceleration of the rolling thing. Let's call it Fc.
Fc R = torque = I alpha
so
alpha = Fc R/I
BUT
alpha is related to a, the linear acceleration down the slope
alpha = a/R
and the gravity force down the slope - Fc = m a
m g sin theta - Fc = m a
m g sin Theta - Fc = m a
m g sin Theta - I a/R^2 = m a
a (m + I/R^2) = m g sin Theta
a = g sin Theta/ (m + I/R^2)
the bigger I/R^2, the lower a
in other words if the mass is at the outer edge, it goes slowly
The hula hoops has all its mass concentrated at the outer radius, it has the slowest a
To rank the objects from fastest average rolling speed to slowest, we can consider their rotational inertia. Rotational inertia is determined by the mass and distribution of mass around the rotation axis, with objects having lower rotational inertia rolling faster.
1. Large marble: The large marble has a relatively small mass, and its mass is well-distributed close to its rotation axis. Hence, it will have the fastest average rolling speed among the listed objects.
2. Penny: The penny also has a small mass, but its mass is evenly distributed throughout its shape. Although slightly slower than the large marble, the penny will have a relatively fast average rolling speed.
3. Basketball: The basketball has a much larger mass compared to the previous two objects. Its mass is distributed away from the rotation axis, which increases its rotational inertia. Consequently, the basketball will have a slower average rolling speed than the large marble and penny.
4. Hula hoop: The hula hoop has the largest mass among the listed objects and also has a significant portion of its mass distributed away from the rotation axis. This makes it have the highest rotational inertia, resulting in the slowest average rolling speed compared to the other objects.
Therefore, the rank from fastest to slowest average rolling speed is as follows:
1. Large marble
2. Penny
3. Basketball
4. Hula hoop
To determine the order from fastest average rolling speed to slowest for the objects listed (penny, large marble, basketball, and hula hoop), we need to consider their properties that affect their rolling speed. The rolling speed of an object depends on the moment of inertia and mass distribution.
1. Moment of inertia: It represents an object's resistance to changes in rotational motion. Objects with smaller moment of inertia tend to roll faster compared to those with larger moment of inertia.
2. Mass distribution: Objects with a higher concentration of mass towards the center roll faster compared to objects with mass distributed away from the center.
Now let's analyze the objects:
- Penny: Pennies have a small moment of inertia since they are thin and have most of their mass concentrated towards the center. Due to their small size and low mass, pennies should roll relatively fast.
- Large marble: Marbles, depending on their size, usually have a similar distribution of mass as pennies. However, since the marble is larger, it might have a slightly larger moment of inertia and roll slightly slower than the penny.
- Basketball: Basketballs are significantly larger and have a more distributed mass compared to pennies and marbles. The larger size and mass distribution away from the center leads to a higher moment of inertia, which will make the basketball roll slower compared to the penny and marble.
- Hula hoop: Hula hoops are large and have the most distributed mass among the objects listed. With mass distributed away from the center, the hula hoop will have the largest moment of inertia and roll the slowest among the given objects.
Based on the above analysis, the objects can be ranked from fastest average rolling speed to slowest as follows:
1. Penny
2. Large marble
3. Basketball
4. Hula hoop