A cylindrical object can roll down an incline, as shown in Figure 1. The incline is slightly less
than one meter in length. A group of students wants to determine the acceleration of the object
while it is rolling down the incline. The students have access to the following equipment.
• A stopwatch, which can measure time intervals up to 999 s with a precision of 0.01 s
• A clock, which can measure time intervals up to 12 hours with a precision of 1 min ute (60 s)
• A meterstick, which can measure lengths up to 1 m with a precision of 1 mm
• A pair of calipers, which can measure lengths up to 10 cm with a precision of 0.05 mm
(a) Assume the object moves with a constant acceleration as it rolls down the incline. Write an
equation that includes acceleration and quantities that can be measured or obtained from
measurements by using the available equipment in the list.

a = 2d / t²

or

a = 2gh / (3d)

Explanation:

One method is to use the equation:

Δx = v₀ t + ½ at²

d = (0) t + ½ at²

d = ½ at²

a = 2d / t²

By measuring the length of the incline d, and the time it takes to reach the bottom t, the students can calculate the acceleration, using only the meter stick and the stopwatch.

Another method is to use conservation of energy to find the final velocity.

Initial potential energy = final rotational energy + kinetic energy

PE = RE + KE

mgh = ½ Iω² + ½ mv²

For a solid cylinder, I = ½ mr².  For rolling without slipping, ω = v/r.

mgh = ½ (½ mr²) (v/r)² + ½ mv²

mgh = ¼ mv² + ½ mv²

mgh = ¾ mv²

4gh/3 = v²

Using constant acceleration equation:

v² = v₀² + 2aΔx

4gh/3 = 0² + 2ad

a = 2gh / (3d)

Using this equation, the students can measure the height of the incline h, and the length of the incline d, to calculate the acceleration.  The only equipment needed is the meter stick.

50 inches should be the resultant vector