To determine which shape will reach the bottom of the incline first, let's analyze the physics involved.

In this situation, each object rolls down the incline without slipping, meaning both rotational and translational kinetic energy need to be considered. The energy conservation equation for a rolling object is:

$mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

where:

• $m$ is the mass,
• $g$ is the acceleration due to gravity,
• $h$ is the height of the incline,
• $v$ is the linear velocity at the bottom,
• $I$ is the moment of inertia,
• $\omega$ is the angular velocity.

Since the objects are rolling without slipping, $v = \omega R$, where $R$ is the radius of the object. Substituting $\omega = \frac{v}{R}$ into the equation and simplifying, we get:

$mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\frac{v^2}{R^2}$

Rearranging this to solve for $v$:

$v^2 = \frac{2mgh}{m + \frac{I}{R^2}}$

This shows that the final speed $v$ depends on the object's moment of inertia $I$. The smaller the moment of inertia relative to $mR^2$, the greater the speed $v$ at the bottom of the incline.

Comparing Moments of Inertia:

1. Sphere: $I_{\text{sphere}} = \frac{2}{5}MR^2$
2. Solid Cylinder: $I_{\text{solid cylinder}} = \frac{1}{2}MR^2$
3. Hollow Cylinder: $I_{\text{hollow cylinder}} = MR^2$

The sphere has the smallest moment of inertia ($\frac{2}{5}MR^2$), followed by the solid cylinder ($\frac{1}{2}MR^2$), and finally the hollow cylinder ($MR^2$).

Conclusion:

Since the sphere has the smallest moment of inertia, it will have the highest linear velocity $v$ at the bottom of the incline and thus will reach the bottom first. Therefore, the sphere should be the shape that reaches the bottom of the incline first in this race.

Would you like more details on the calculations, or do you have any questions?

Here are some related questions for further exploration:

1. How does the angle of the incline affect the outcome of the race?
2. What happens if friction is introduced between the objects and the incline?
3. How would the results differ if the objects started with different initial velocities?
4. How would a different mass distribution affect the moments of inertia?
5. What if we used an incline with variable slope – how would that influence the race?

Tip: The object with the smallest moment of inertia for a given mass and radius will typically accelerate faster when rolling, as more of its energy goes into translational (linear) motion rather than rotational.