The five masses below all have the same radius and a cylindrically symmetric mass distribution. They start to roll down an inclined plane, starting from rest, at the same time and from the same height. Give their order of arrival at the bottom.
(a) Icm=935 g*cm2, M = 44g
(b) Icm=1046 g*cm2, M = 50g
(c) Icm=1232 g*cm2, M = 53g
(d) Icm=755 g*cm2, M = 53g
(e) Icm=975 g*cm2, M = 47g
To determine the order of arrival at the bottom of the inclined plane for the given masses, we need to compare their moments of inertia (Icm) and their masses (M). The mass with the smallest moment of inertia and/or the largest mass will arrive first at the bottom.
Let's compare the values for each mass:
(a) Icm = 935 g*cm^2, M = 44 g
(b) Icm = 1046 g*cm^2, M = 50 g
(c) Icm = 1232 g*cm^2, M = 53 g
(d) Icm = 755 g*cm^2, M = 53 g
(e) Icm = 975 g*cm^2, M = 47 g
Comparing the moments of inertia, we see that mass (d) has the smallest Icm at 755 g*cm^2. Next is mass (a) with Icm = 935 g*cm^2, followed by mass (e) with Icm = 975 g*cm^2. The largest moments of inertia are for masses (b) and (c), with Icm = 1046 g*cm^2 and Icm = 1232 g*cm^2, respectively.
Now, let's compare the masses. Mass (c) has the largest mass at 53 g, followed by masses (d) and (e) at 53 g and 47 g, respectively. Masses (b) and (a) have the smallest masses at 50 g and 44 g, respectively.
Based on the comparison, mass (d) has the smallest Icm and the same mass as mass (c), so mass (d) will arrive first at the bottom of the inclined plane.
The order of arrival at the bottom from earliest to latest is:
1. Mass (d)
2. Mass (c)
3. Mass (e)
4. Mass (b)
5. Mass (a)
To determine the order of arrival of the five masses at the bottom of the inclined plane, we need to consider their moments of inertia relative to the rotational axis.
The formula for the rotational kinetic energy of a rolling object is given by:
KE = (1/2) * I * ω^2
Where:
- I is the moment of inertia of the object
- ω is the angular velocity of the object
When the objects start to roll down the inclined plane, their potential energy will be converted to both translational and rotational kinetic energy. For these objects, the total kinetic energy will be the sum of their translational and rotational kinetic energies.
If the objects have the same radius and start from the same height, their potential energy will be the same. Therefore, the object with the lowest total kinetic energy (translational + rotational) will reach the bottom first.
To determine the order, we can calculate the total kinetic energy for each object using the following steps:
Step 1: Calculate the translational kinetic energy
The translational kinetic energy can be calculated using the formula:
KE_trans = (1/2) * M * v^2
Where:
- M is the mass of the object
- v is the velocity of the object
Since all the masses start from rest and have the same height, their velocities will be the same. Therefore, we can compare their translational kinetic energies by comparing their masses (M).
Step 2: Calculate the rotational kinetic energy
The rotational kinetic energy can be calculated using the formula mentioned earlier:
KE_rot = (1/2) * I * ω^2
Where:
- I is the moment of inertia of the object
- ω is the angular velocity of the object
Since the objects start from rest, their angular velocities will be equal. We can compare their rotational kinetic energies by comparing their moments of inertia (I).
Step 3: Calculate the total kinetic energy
The total kinetic energy is the sum of the translational and rotational kinetic energies:
KE_total = KE_trans + KE_rot
By comparing the total kinetic energies for each object, we can determine their order of arrival at the bottom. The object with the lowest total kinetic energy will arrive first, and the one with the highest total kinetic energy will arrive last.
Now, let's calculate the total kinetic energies for each object and determine their order of arrival at the bottom of the inclined plane:
(a) Icm = 935 g*cm^2, M = 44 g
(b) Icm = 1046 g*cm^2, M = 50 g
(c) Icm = 1232 g*cm^2, M = 53 g
(d) Icm = 755 g*cm^2, M = 53 g
(e) Icm = 975 g*cm^2, M = 47 g
To perform the calculations, we need to convert the moments of inertia from g*cm^2 to kg*m^2. We also need to convert the masses from grams to kilograms.
Icm(a) = 935 g*cm^2 = 935 * 10^-7 kg*m^2
M(a) = 44 g = 44 * 10^-3 kg
Icm(b) = 1046 g*cm^2 = 1046 * 10^-7 kg*m^2
M(b) = 50 g = 50 * 10^-3 kg
Icm(c) = 1232 g*cm^2 = 1232 * 10^-7 kg*m^2
M(c) = 53 g = 53 * 10^-3 kg
Icm(d) = 755 g*cm^2 = 755 * 10^-7 kg*m^2
M(d) = 53 g = 53 * 10^-3 kg
Icm(e) = 975 g*cm^2 = 975 * 10^-7 kg*m^2
M(e) = 47 g = 47 * 10^-3 kg
Now, we can calculate the total kinetic energies using the formulas mentioned earlier:
TotalKineticEnergy(a) = KE_trans(a) + KE_rot(a)
TotalKineticEnergy(b) = KE_trans(b) + KE_rot(b)
TotalKineticEnergy(c) = KE_trans(c) + KE_rot(c)
TotalKineticEnergy(d) = KE_trans(d) + KE_rot(d)
TotalKineticEnergy(e) = KE_trans(e) + KE_rot(e)
Comparing the total kinetic energies of the objects will provide us with the order of arrival at the bottom of the inclined plane.