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 0.01s
• A clock, which can measure time intervals up to 12 hours with a precision of 1 minute (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
(1) 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.

When carrying out experiment to find the acceleration, it is important that the experiment should be repeated several times when using the stopwatch

The acceleration of the cylindrical object down the plane can be measured using the meterstick , the stopwatch and the following equation with several repeated measurements;

With the use of only the meterstick, the height, h, and length, d, of the ramp can be measured, from which the acceleration can be found with the equation;

The reason the above equations can be used to find the acceleration of the cylinder are as follows:

The available equipment are;

A stop watch with time intervals up to 999 seconds and a precision of 0.01 seconds A 12 hour clock with 1 minute precisionA meterstick that can measure up to a meter with 1 mm precisionA pair of calipers with capacity to measure up to 10 cm and having a precision of 0.05 mm

The equations of constant acceleration are;

1)

2)

3)

From the first equation, the initial velocity is zero, therefore, the acceleration becomes;

Where:

d = The length of the ramp, measured with the meter rule

t = The time it takes the cylinder to reach the bottom of the ramp, measured with the stopwatch

From equation (2), the acceleration is given as follows;

v² = 2·a·d

Which can be measured using the stopwatch and the meter rule

Energy conservation method

Using the energy conservation principle, gives;

Potential energy = Total kinetic energy

Potential energy = m·g·h

Total kinetic energy = Translational inertia + Rotational inertia

Total kinetic energy =

For the solid cylinder, we get;

Therefore;

Therefore, the height, h and the length of the ramp, d, can be measured with the meter rule, from which the acceleration can be determined

Learn more about experiments to determine acceleration here:

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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.

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