In a series of successive measurements in an experiment, the readings of the period of oscillation of simple pendulum were found to be 3.0s, 3.1s,3.2s,3.3s,3.4s and 3.5s Calculate The mean Value, mean absolute error, Relative error , Percentage error (%)
Answers
Answer:
We have now seen how to calculate the average velocity between two positions. However, since objects in the real world move continuously through space and time, we would like to find the velocity of an object at any single point. We can find the velocity of the object anywhere along its path by using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later chapters.
Instantaneous Velocity
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply velocity. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero. To illustrate this idea mathematically, we need to express position x as a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is
\[\overset{\text{-}}{v}=\frac{x({t}_{2})-x({t}_{1})}{{t}_{2}-{t}_{1}}\]
. To find the instantaneous velocity at any position, we let
\[{t}_{1}=t\]
and
\[{t}_{2}=t+\text{Δ}t\]
. After inserting these expressions into the equation for the average velocity and taking the limit as
\[\text{Δ}t\to 0\]
, we find the expression for the instantaneous velocity:
\[v(t)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{x(t+\text{Δ}t)-x(t)}{\text{Δ}t}=\frac{dx(t)}{dt}.\]
Instantaneous Velocity
The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of x with respect to t:
\[v(t)=\frac{d}{dt}x(t).\]
Like average velocity, instantaneous velocity is a vector with dimension of length per time. The instantaneous velocity at a specific time point
\[{t}_{0}\]
is the rate of change of the position function, which is the slope of the position function
\[x(t)\]
at
\[{t}_{0}\]
. (Figure) shows how the average velocity
\[\overset{\text{-}}{v}=\frac{\text{Δ}x}{\text{Δ}t}\]
between two times approaches the instantaneous velocity at
\[{t}_{0}.\]
The instantaneous velocity is shown at time
\[{t}_{0}\]
, which happens to be at the maximum of the position function. The slope of the position graph is zero at this point, and thus the instantaneous velocity is zero. At other times,
\[{t}_{1},{t}_{2}\]
, and so on, the instantaneous velocity is not zero because the slope of the position graph would be positive or negative. If the position function had a minimum, the slope of the position graph would also be zero, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity function give the minimum and maximum of the position function.
Graph shows position plotted versus time. Position increases from t1 to t2 and reaches maximum at t0. It decreases to at and continues to decrease at t4. The slope of the tangent line at t0 is indicated as the instantaneous velocity.
Figure 3.6 In a graph of position versus time, the instantaneous velocity is the slope of the tangent line at a given point. The average velocities
\[\overset{\text{-}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{{x}_{\text{f}}-{x}_{\text{i}}}{{t}_{\text{f}}-{t}_{\text{i}}}\]
between times
\[\text{Δ}t={t}_{6}-{t}_{1},\text{Δ}t={t}_{5}-{t}_{2},\text{and}\,\text{Δ}t={t}_{4}-{t}_{3}\]
are shown. When
\[\text{Δ}t\to 0\]
, the average velocity approaches the instantaneous velocity at
\[t={t}_{0}\]
.