Final answer:
To determine how much sooner a box reaches the bottom of an incline than a disk, physics concepts such as potential energy, kinetic energy, and rotational inertia must be taken into account. The disk takes longer due to its rotational kinetic energy. Calculations would involve the conservation of mechanical energy and the balancing of forces on an incline.
Step-by-step explanation:
The question you've posed involves understanding the motion of objects down an incline and comparing the time it takes for different objects with various motions (a rolling disk versus a sliding box) to travel the same distance. To determine how much sooner the box reaches the bottom of the incline than the disk, you'll need to consider concepts like kinetic energy, potential energy, and the rotational inertia of objects. In physics, these concepts are used to explain why objects move differently under similar conditions.
For part (b) of question 31, you're asked to find the ratio of the time the rolling cylinder takes to reach the height on the second incline to the time the sliding cylinder takes. This requires an understanding of the conservation of mechanical energy and the moment of inertia for the rolling object. Since energy conservation dictates that they reach the same height, the difference in time is due to the rotational kinetic energy that the rolling cylinder has, which the sliding cylinder does not. The disk's rotational inertia means it will take longer to reach the bottom of the incline due to the additional energy stored in rotation.
For (c), explaining why the time for rolling motion is greater than that for sliding motion, we look to the rotational kinetic energy again. The rolling cylinder has to use some of its potential energy to cause rotation, which means less energy is converted to translational motion compared to the sliding cylinder that converts all its potential energy into translation.
Example Calculation
- The moment of inertia of the rolling object affects its final velocity at the bottom of the incline, with more rotational inertia resulting in a slower speed for the same height descended.
- Finding the angle of the slope that allows a box to move at a constant velocity involves balancing the forces parallel and perpendicular to the incline with the force of friction.