The experiment explores linear motion under constant acceleration to express displacement, acceleration and velocity as functions of time. In addition, it explores the constant motion under an inclined position. The motion of objects can be described by kinematics equations. This is a set of equations that fully describe the relationships among the velocity, displacement, acceleration, and position of object under linear motion (Serway and Jewett 35). These kinematics equations for constant acceleration are shown below:
X=X0 + V0t + 1/2at2
Vf = V0 + at
V2f = V2 0 + 2a(X-X0)
Where X= final position
X0 = initial position
V0 = Initial velocity
Vf =final velocity
a = acceleration (Resnick, Halliday, and Krane 28).
The instantaneous velocity = v=dx/dt
Instantaneous acceleration =a =dv/dt
Average velocity = Vave =∆V/∆t
Average velocity = aave = ∆X/∆t
The motion down an incline is linear; thus, with constant acceleration. The force F is dependent on the mass of the body and the angle of inclination, while the acceleration of the object down the incline is not dependent on mass (Serway, Faughn and Vuille 103). However, it depends on the angle of inclination. Therefore, F is proportional to the mass of the body.
Considering the forces acting on the object on motion as shown below
Assuming that the surface is frictionless and the only force acting on the body is due to its weight. Then, -T + Mgsinθ = 0 (Serway, Faughn and Vuille 103).
T = Mgsinθ, but. F = Ma
= Ma = Mgsinθ, therefore, a = gsinθ ((Resnick, Halliday, and Krane 32).
Experimental Methodology
Motion on incline
The materials required for the experiment included a force sensor, track, cart, force sensor, angle indicator, height adjuster, foam, bumper, ring stand, and balance. Initially, the experiment is set as shown below.
The force and acceleration the cart experience down the incline were measured as a function of the angle of inclination. A string was attached to the mass on one end and to the force detector on the other end. At this point, the force was measured at a stationary position of the cart. The cart was released, and the position, velocity and acceleration of the cart measured with time. The velocity, position and acceleration time functions were measured for different values of angle θ: 5, 11, 14 and 17 degrees.
Motion Constant acceleration
The angle of inclination was set at 11 degrees. In addition, the motion of the cart was measured as a function of velocity and position against time. A set of values for velocity against time, and position against time were obtained. Moreover, three consecutive and non consecutive positions for X were obtained and used to calculate the instantaneous velocity, acceleration and to test the kinematic equations.
Data Analysis And Discussions
- Inclined Motion
- 17 degrees
- 14 degrees
- 11degrees
- 5 degrees
The graph of Velocity against time for angle θ=5o
The graph of velocity against time is linear indicating a constant acceleration for the motion. Therefore, the measured acceleration a=0.679m/s2.
The graph of Velocity against time for angle θ=11o
The graph of velocity against time is linear indicating a constant acceleration for the motion. Therefore, the measured acceleration a=1.776m/s2.
The graph of Velocity against time for angle θ=14o
The graph of velocity against time is linear indicating a constant acceleration for the motion. Therefore, the measured acceleration a=2.164m/s2.
The graph of Velocity against time for angle θ=17o
The graph of velocity against time is linear indicating a constant acceleration for the motion. Therefore, the measured acceleration a=2.676m/s2.
Using g=9.813m/s2
The results show variations in measured values of force and acceleration from the theoretical values. In this case, the measured acceleration is less than the theoretical values. These phenomena can be attributed to experimental assumptions that the cart used is run on a completely frictionless surface and the total external force acting on the body is due to gravity. However, this may not be the case. In addition, variations in the force measured and theoretical values indicate that the measured force is greater than the theoretical values calculated. Again, experimental errors when releasing the cart can account for this external force. This is because additional force can be exerted at the time of release of the cart; thus, interfering with the motion of the object. In an ideal, the acceleration is independent of the force F; however, the values of the measured force and acceleration should be the same.
The graph of force against acceleration
The graph of measured force against acceleration is linear. The slope of the graph is equal to 0.619. The relationship of mass and acceleration shows that the slope represents the mass of the object. Therefore, the mass is 0.645kg. This value is compared to the actual mass of 0.5698kg, though is slightly higher. This can be attributed to the above mentioned possible errors in force and acceleration; since, the force measured was higher than the theoretical value and the acceleration lower than the theoretical value. Inferring from the relationship F = Ma, increasing F and reducing acceleration “a” would result to higher value of M.
Motion Under constant acceleration
The angle is set at 11oand the mean acceleration is 1.675m/s2
Using θ=110 for the study of constant acceleration.
The graph of velocity verses time for 11o
11degrees
The graph of position against time for 11o
The value of A=0.836; this is comparable to 0.888value which is half of the acceleration from the graph of velocity against time of 1.776m/s2. The acceleration of 1.776m/s2 is comparable to the instantaneous acceleration mean acceleration value of 1.675m/s2.
X= 0.836t2 - 3.053t + 3.054
Calculating the position at a time t = 2.2799s, the value of X is obtained as follows.
X=0.836 *2.27992 -3.053*2.2799 +3.054 = 0.4389m, this is comparable to 0.4378 m obtained above when applying the kinematics equation.
Comparing the equation of displacement dependency: X= 0.836t2 - 3.053t + 3.054 with the equation obtained above in testing equations kinematics: X=0.3703 + 0.5918t +0.886t2, it is clear that the constant A has the same value, while B and C varies. This is because the acceleration in the motion is constant, while velocity changes.
The function that describes the position verses time is known as displacement function. It indicates the position of a moving object at a time t. the comparison of the equations of displacement both for kinematic equations and the equation showing quadratic dependency of the displacement would yield similar results. However, the acceleration would be large; since the angle of inclination would be increase and Acceleration is dependent on force which increases with the sinθ.
The acceleration is independent of the mass of the body; therefore, it depends on sinθ. The acceleration will be at a maximum value at sinθ.=1. Therefore, acceleration given by g sinθ will be equal to g=9.813m.s2 at its maximum value.
An object falling in a frictionless inclined surface experiences a constant acceleration. In addition, the force is directional proportional to the acceleration. However, the force is dependent on mass of the body, while the acceleration is not. The motion of constant acceleration indicates that the equations of kinematics give the same results as velocity and position time graphs; thus, they can be used to represent and describe motion.
Resnick, Robert, Halliday David, and Krane Kenneth. Physics, Volume 1. New York : Wiley,
2002. Print.
Serway, Raymond, Faughn S. Jerry, and Vuille Chris. College Physics, Volume 1. Canada:
Cengage Learning, 2011. Print.
Serway, Raymond, and Jewett W. John. Physics for scientist and engineers, Volume 1. Canada:
Cengage Learning, 2009. Print.
