Investigating the relationships of angle at which the ramp is inclined and the acceleration that the object experiences while moving down along the ramp

Title

Investigating the relationships of angle at which the ramp is inclined and the acceleration that the object experiences while         moving down along the ramp.

Research question

What is the relationship between the angle of fixed-length track inclination and the acceleration of  the metal sphere rolling down the track from rest?

The objective of this investigation is to determine how the incline of the constant-length ramp influences the acceleration of the metal sphere.

By systematically adjusting the vertical height of one end of the ramp while the total track length remains unchanged, the inclination angle  is modified according to the geometric properties of a right triangle. During each trial, the metal ball is released from the highest point of the ramp, and a stopwatch is used to record the time taken for it to reach the base. This measured time, combined with the known displacement along the ramp, allows for the calculation of the ball's acceleration in .[pic 1][pic 2]

To evaluate the accuracy of the experiment and identify potential technical errors, the calculated acceleration and the sine of the angle will be used to derive an experimental value for the acceleration due to gravity. This final result will then be compared to the standard accepted value of g to determine the overall precision and percentage of experimental error.

Background information

When an object of mass m in kilogram (kg) is placed on an inclined plane at angle , the force of gravity mg in meter per second squared () acts downward. However, since the force of gravity is not perpendicular to the inclined surface, it is divided into two components: mgsin is the parallel to the plane and mgcos is perpendicular to the plane. Since the parallel component is of the gravitational force causes the object to accelerate, it can be written as[pic 3][pic 4][pic 5][pic 6]

F = mgsin[pic 7]

Since the friction is assumed to be negligible, this force is the net force that causes the motion of the object. Therefore, the object (sphere) experiences only translational motion.

According to Newton’s second law of motion,

F = ma

Hence,

ma = mgsin[pic 8]

a = gsin[pic 9]

Here, a is the acceleration of the sphere in meters per squared second (), g is the acceleration of free fall on Earth (), and  is the angle of inclination of the track. [pic 10][pic 11][pic 12]

In order to find the acceleration of the object, the following equation can be used

s = ut + 0.5  a[pic 13][pic 14]

Where s is the displacement along the ramp in meters (m), u is initial velocity in meter per second (), t is time taken in seconds (s) and a is the object’s acceleration in meter per second square ().[pic 15][pic 16]

Since the initial velocity (u) of the object is zero the formula can be rewritten as:

s = 0.5  a[pic 17][pic 18]

Therefore,

a =  [pic 19]

Where the length of the ramp s is constant, and time taken for sphere to travel along the ramp t will be measured during the experiment. As a result, acceleration of the object could be calculated.

In this case, sin can be calculated by the following trigonometric formula:[pic 20]

sin = [pic 21][pic 22]

The opposite cathetus equals the height h of the one end of the ramp (where the sphere is placed), and the hypotenuse is the length s of the ramp. Therefore, the equation can be rewritten as:

sin = [pic 23][pic 24]

The length of the ramp is constant, meaning that any change in height will change sin as well, thus affecting the acceleration of the sphere. [pic 25]

In relation to the research question, the relationship between the inclination angle of the track and the acceleration of the sphere can be represented as in the Figure 1.

[pic 26]

Fig 1. Expected graph of acceleration against inclination angle

The graph shows a non-linear relationship between acceleration and inclination angle. When the angle increases, acceleration of the metal sphere increases as well. Although the acceleration stops to increase when it reaches the value of g or approximately 9.8 .[pic 27]

Since a = gsin, a  sin. It means that acceleration of the sphere in meter per squared second is proportional to the sine of the inclination angle. The linearized graph could be drawn as in the Figure 2. [pic 28][pic 29][pic 30]

[pic 31]

Fig 2. Expected graph of acceleration against sin[pic 32]

The gradient of the line can be calculated by the formula below

gradient =  =  = g[pic 33][pic 34]

It means that the gradient of the linearized graph is equal to the acceleration of free fall.

The experimental error will be calculated with the reference to 9.8  as the accepted value by the following formula:[pic 35]

experimental error =   100%[pic 36][pic 37]

Variables

Independent variable

The independent variable in this experiment is the angle of inclination (°) of the ramp. Although the exact angle values are not required, it is essential that the angle changes. This will be achieved by increasing the height of one end of the ramp using an adjustable stand, while the other end remains fixed. The time will be measured across the following ranges: 6 cm, 9 cm, 12 cm, 15 cm, and 18 cm, increasing in intervals of 3 cm. The height will be measured from the ground to the raised end of the ramp using an analog measuring tape with a smallest division of 0.1 cm, resulting in an uncertainty of ± 0.05 cm.