A simple pendulum is oscillating without damping. When the displacement of the bob is less than maxi

Q: Why isn't the acceleration just pointing straight down due to gravity?

While gravity acts straight down, the string constrains the bob to move in a circular arc. The string exerts a tension force that alters the net force, and thus the net acceleration, making it point inwards rather than straight down.

Q: Why is there a radial acceleration if the bob is just swinging?

Any object moving along a curved path must have a component of acceleration directed towards the center of curvature. This is the centripetal acceleration ($\frac{v^2}{L}$), which is responsible for changing the direction of the velocity vector.

Q: Does the net acceleration ever point exactly along the tangent?

Yes, but only at the extreme positions (maximum displacement). At the extremes, the velocity is momentarily zero, so the radial acceleration ($\frac{v^2}{L}$) is zero. The only acceleration left is the tangential one.

Q: Does the net acceleration ever point exactly along the string?

Yes, at the lowest point (mean position). Here, the tangential acceleration ($g \sin\theta$) is zero because $\theta = 0$. The only acceleration is the radial one, pointing straight up towards the support.

Q: Why can't the acceleration point outwards away from the pivot?

The radial component of acceleration must always point towards the center of the circular path to keep the object turning. If it pointed outwards, the bob would be breaking the string and flying away!

Question

A simple pendulum is oscillating without damping. When the displacement of the bob is less than maximum, its acceleration vector $\mathbf{a}$ is correctly shown in

Options:

(A) Option (a)

(B) Option (b)

(C) Option (c)

(D) Option (d)

Sigma Percentile · Accepted Answer

### The Deceptive Simplicity of the Pendulum

A simple pendulum is the quintessential physics system. It looks so peaceful, just swinging back and forth. But beneath this tranquil exterior lies a beautiful, dynamic interplay of forces and accelerations.

When we observe the bob at an intermediate position—neither at the very top of its swing nor at the very bottom—it is in a state of **non-uniform circular motion**. It is moving along a circular arc, and its speed is constantly changing.

### The Two Faces of Acceleration

Because the motion is non-uniform and circular, the total acceleration of the bob cannot be described by a single, simple vector pointing in an obvious direction. Instead, it is the vector sum of two distinct components.

We must break the acceleration down into a **tangential component** and a **radial component**.

### The Tangential Pull

First, let's consider why the bob speeds up as it falls towards the center. Gravity is constantly pulling the bob straight down.

However, the string restricts the bob's path. A component of this gravitational force acts along the tangent to the circular path. This tangential force is what accelerates the bob along the arc.

The tangential acceleration is given by:
$$a_t = g \sin	heta$$

This vector, $\mathbf{a}_t$, always points along the tangent, directed towards the lowest point (the mean position).

### The Radial Tug

Next, we must account for the fact that the bob is moving along a curve. Even if its speed were constant, changing direction requires an acceleration.

Since the bob has some velocity $v$ at this intermediate position, it experiences a **centripetal** or **radial acceleration**. This acceleration is responsible for continuously bending the bob's path into a circle.

The magnitude of this radial acceleration is:
$$a_n = \frac{v^2}{L}$$

This vector, $\mathbf{a}_n$, is directed strictly along the string, pointing straight towards the point of suspension.

### The Grand Synthesis

We now have two acceleration vectors acting on the bob simultaneously: $\mathbf{a}_t$ pulling it tangentially, and $\mathbf{a}_n$ pulling it radially inwards.

The true, net acceleration $\mathbf{a}$ is the vector sum of these two components:
$$\mathbf{a} = \mathbf{a}_t + \mathbf{a}_n$$

By applying the parallelogram law of vector addition, we can visualize this resultant vector. Since both components are non-zero and perpendicular to each other, the resultant vector must lie diagonally between them.

It points inwards and upwards relative to the tangent line. When we examine the given options, only one diagram correctly captures this physical reality. The vector in option (c) perfectly depicts the net acceleration pointing between the string and the tangent.

This elegant vector addition reveals the hidden complexity of a simple swinging bob!

Animated Solution for Physics - Kinematics: A simple pendulum is oscillating without damping. When the displacement of the bob is less than maximum, its acceleration vector $a$ is correctly shown in

Select Answer:

Visualized Solution

The Pendulum Setup

State of the Bob

Acceleration in Circular Motion

Tangential Acceleration ( $a_{t}$ )

Radial Acceleration ( $a_{n}$ )

Net Acceleration Vector

Resultant Direction

The Sigma Insight: Kinematics of Circular Motion

The Deceptive Simplicity of the Pendulum

The Two Faces of Acceleration

The Tangential Pull

The Radial Tug

The Grand Synthesis

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