How Does Displacement Change Over Time in Periodic Motion?
Let the initial position of a particle be x0 and position at time t be x. Then its displacement relative to x0 is x – x0 and it depends in some manner on time t. We say that displacement varies with time or is a function of time t.
We denote the function by ft. Thus x – x0 = ft ,
Here the expression for function depends upon type of motion. If the motion is with uniform velocity we have x – x 0 = Vt linear function Here we say that displacement x – x0 is proportional to t. Position x = x0 + Vt is said to be a linear function of t as power oft is 1. The graph of xt is a straight line.
If the motion is along the x-axis with uniform acceleration, the displacement is given by x – x0 = ut + ½ 2 at 2.
Here we have quadratic function oft as the highest power of is 2. The graph of xt is a parabola. If acceleration is not uniform we have an infinite number of ways in which acceleration can change. Of these, one special case is very important-arising from a periodic function of time.
Derivation of Displacement as a Function of Time
In order to understand displacement as a function of time, we will have to derive an expression for displacement also known as the Second equation of motion.
Let us assume a body traveling with an initial velocity of v1 at time t1, and is subjected to some constant accelerations thus making its final velocity of v2 at time t2,
Keeping these things in assumption let’s derive the following.
We know that average velocity is equal to total displacement covered in a given time interval. Thus using this we can say that
V average = Total displacement / Total time
Displacement = V average Δt
Where Δt is the change in time and is equal to t2-t1
Since acceleration is constant thus average velocity is mean of initial and final velocity
Therefore ,displacement = (V1+ V2/2) Δt , where V1 and V2 are initial and final velocity
Now, since acceleration is constant, final velocity
V2 = V1 +at
d = ((V1+V1 + aΔt) /2 ) Δt
Now we can if we rewrite the above as,
d = (2V1+ aΔt) Δt /2
The above expression is second equation of motion and is one of the most fundamental expressions in kinematics and is finally reduced to
d= V1 t+½ at2
Where V1 is the initial velocity and t is the change in time, all the quantities in this derivation like Velocity, displacement and acceleration are vector quantities.
Thus, the above expression clearly proves that displacement depends upon the time .
Example of an Oscillating Pendulum
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When Bob reaches the highest point, the potential energy is maximum and the kinetic energy is minimum as the velocity is equal to zero but the total energy is conserved throughout the motion and only transformation from kinetic energy to potential energy or vice versa will take place.
Thus we can easily deduce that the velocity is equal to zero, by looking at the slope of the displacement-time graph at specific times.
The Slope of the graph at A is positive stating that the velocity of the body is also positive, or in the forward, direction while Slope at B is equal to zero meaning that the velocity of the body is zero, while acceleration is still there that later imparts velocity in reverse direction.Thus slope at C is negative in the graph means that the velocity of the body is also negative, or in the reverse direction.
If we draw a displacement-time graph of this oscillating pendulum we will get something like this as shown below in the figure. Here the magnitude of velocity is always positive, it is the direction that decides if the velocity is positive or negative.
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The displacement of the bob of the pendulum is periodic in nature as it repeats itself after a certain amount of time. Also the displacement at any given time in the future can be predicted with the help of graph since we know the time and the time period of the pendulum.
Thus we can say that displacement of the oscillating pendulum bob is a function of time.
FAQs on Displacement as a Function of Time and Periodic Functions
1. What is meant by representing displacement as a function of time in Physics?
Representing displacement as a function of time means using a mathematical equation, typically written as x(t), to describe an object's position (displacement) at any specific moment in time (t). This function provides a complete description of the object's path, allowing us to calculate its position, velocity, and acceleration at any instant without needing to observe it continuously.
2. What is the general mathematical formula for displacement as a function of time in periodic motion?
Any motion that repeats itself at regular intervals is periodic. The displacement in such motion can be described by a periodic function. The most fundamental periodic functions used are sine and cosine. Therefore, the displacement `x(t)` can often be represented as x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
3. How is displacement represented as a function of time specifically for Simple Harmonic Motion (SHM)?
Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force is directly proportional to the displacement. The displacement `x` from the mean position at any time `t` is precisely described by the sinusoidal function: x(t) = A cos(ωt + φ). This equation is the hallmark of SHM and models systems like an ideal mass on a spring or a simple pendulum with small oscillations.
4. In the SHM equation x(t) = A cos(ωt + φ), what is the physical significance of amplitude (A), angular frequency (ω), and phase constant (φ)?
Each term in the SHM displacement function has a distinct physical meaning:
- Amplitude (A): This represents the maximum displacement or distance the object moves from its equilibrium (mean) position. It defines the boundaries of the motion.
- Angular Frequency (ω): This determines how rapidly the oscillations occur. It is related to the time period T by the formula ω = 2π/T. A higher ω value means more oscillations per second.
- Phase Constant (φ): Also known as the initial phase angle, this constant specifies the object's position and direction of motion at the starting time, t=0. It essentially shifts the start of the wave in time.
5. Why is the displacement-time graph for an oscillating object a curve, and what does its slope signify?
The displacement-time graph for an oscillating object (like a pendulum) is a sinusoidal curve (sine or cosine wave) because its acceleration is not constant; it changes continuously, being maximum at the extremes and zero at the mean position. The slope of this graph at any point represents the object's instantaneous velocity. The slope is steepest at the mean position (maximum velocity) and becomes zero at the extreme positions, where the object momentarily stops before reversing direction.
6. How does the displacement function for periodic motion differ from the displacement equation for constant acceleration?
This is a critical distinction. The displacement function for periodic motion, like x(t) = A cos(ωt), describes an object whose acceleration is constantly changing. In contrast, the kinematic equation s = ut + ½at² is only valid when the acceleration 'a' is constant, such as in free fall. Applying the constant acceleration formula to an oscillator would give incorrect results because the oscillator's acceleration is a function of its position.
7. What are some real-world examples of displacement being a periodic function of time?
Several real-world phenomena can be modelled using displacement as a periodic function of time. Key examples include:
- The to-and-fro motion of a simple pendulum (for small angles).
- The vertical or horizontal oscillation of a mass attached to a spring.
- The vibration of a tuning fork's prongs after being struck.
- The rise and fall of tides in the ocean.
- The position of a spot on a rotating wheel relative to a fixed axis.
