Which of the following statements is false for a particle moving in a circle with a constant angular

Q: If the speed is constant, shouldn't the acceleration be zero?

Velocity is a vector, meaning it has both magnitude (speed) and direction. Even if the speed is constant, the direction of the particle is constantly changing as it moves in a circle. This change in direction requires an acceleration, specifically centripetal acceleration.

Q: What is tangential acceleration and why is it zero here?

Tangential acceleration is the component of acceleration that changes the magnitude of the velocity (the speed). Since the problem specifies a 'constant angular speed', the linear speed is also constant, meaning the tangential acceleration must be zero.

Q: Can the acceleration vector ever be tangent to the circle?

Yes, but only if the particle is moving in a straight line, which contradicts circular motion! In non-uniform circular motion (where speed is changing), there is a tangential component of acceleration, but the *net* acceleration vector will be at an angle, not purely tangent.

Q: Why are velocity and acceleration perpendicular in this case?

Velocity is always tangential to the path. In uniform circular motion, the only acceleration is centripetal, which is purely radial (pointing to the center). Geometrically, a radius and a tangent at the same point on a circle are always perpendicular.

Q: What happens if the angular speed is not constant?

If angular speed varies, the particle experiences both centripetal acceleration (changing direction) and tangential acceleration (changing speed). The net acceleration vector would be the vector sum of these two, pointing somewhere inside the circle but not directly at the center.

Question

Which of the following statements is false for a particle moving in a circle with a constant angular speed?

Options:

(A) The velocity vector is tangent to the circle

(B) The acceleration vector is tangent to the circle

(C) The acceleration vector points to the centre of the circle

(D) The velocity and acceleration vectors are perpendicular to each other

Sigma Percentile · Accepted Answer

### The Anatomy of Uniform Circular Motion

Imagine a particle whizzing around a circular track. The problem gives us a crucial piece of information: the particle is moving with a **constant angular speed** ($\omega$). This specific scenario is known as Uniform Circular Motion (UCM). Let's break down the physical vectors involved in this motion to find which statement is false.

### The Velocity Vector: Always on the Edge

By the fundamental definition of kinematics, the instantaneous velocity vector $\mathbf{v}$ of any moving particle is always directed along the tangent to its path at that specific point.

Whether the particle is speeding up, slowing down, or maintaining a constant speed, its velocity vector will always graze the edge of the circle. Therefore, the statement "The velocity vector is tangent to the circle" is absolutely true.

### The Acceleration Vector: The Center-Seeking Force

Now, let's analyze the acceleration. Acceleration is the rate of change of velocity. Since velocity is a vector, it can change in two ways: a change in magnitude (speed) or a change in direction.

Because our particle has a constant angular speed, its linear speed $v = \omega r$ is also constant. This means there is zero tangential acceleration ($a_t = 0$). There is no force pulling the particle forward or pushing it backward along the track.

However, the *direction* of the velocity is continuously changing as the particle navigates the curve. This continuous change in direction requires an acceleration. This is the **centripetal acceleration** ($\mathbf{a}_c$), and it is always directed radially inward, pointing straight towards the center of the circle.

Therefore, the statement "The acceleration vector points to the centre of the circle" is true.

### The Geometric Relationship

Let's look at the geometry of our two vectors. At any given point on the circle, the velocity vector lies along the tangent line, and the acceleration vector lies along the radial line.

From basic geometry, we know that a tangent to a circle is always perpendicular to the radius at the point of tangency. Consequently, the velocity vector $\mathbf{v}$ and the centripetal acceleration vector $\mathbf{a}_c$ must be perpendicular to each other.

This makes the statement "The velocity and acceleration vectors are perpendicular to each other" true.

### The False Claim

We are left with option (b): "The acceleration vector is tangent to the circle." As we just established, in uniform circular motion, the only acceleration present is the centripetal acceleration, which is purely radial. There is no tangential component.

Thus, claiming the acceleration vector is tangent to the circle is a fundamental misunderstanding of uniform circular motion. This is our false statement.

Animated Solution for Physics - Kinematics: Which of the following statements is false for a particle moving in a circle with a constant angular speed?

Select Answer:

Visualized Solution

\text{Uniform Circular Motion}

\text{Velocity Vector}

\text{Acceleration in UCM}

\text{Centripetal Acceleration}

\text{Relationship between } \mathbf{v} \text{ and } \mathbf{a}

\text{Evaluating Options}

\text{Non-Uniform Circular Motion}

The Sigma Insight: Kinematics of Circular Motion

The Anatomy of Uniform Circular Motion

The Velocity Vector

The Acceleration Vector

The Geometric Relationship

The False Claim

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