Understanding Particle Motion in a Circle with Uniform Speed

The question asks us to identify which quantity remains constant when a particle moves in a circle at a uniform speed. Uniform speed means the magnitude of the particle's velocity does not change. However, motion in a circle involves continuous change in direction. Let's analyze each option.

Analyzing Velocity and Momentum Vectors

We need to consider both the magnitude and direction of vector quantities.

• Instantaneous Linear Velocity Vector: The linear velocity ($\vec{v}$) of a particle is a vector. Its magnitude is the speed, which is given as uniform (constant). However, in circular motion, the direction of the velocity vector is always tangent to the circle at the particle's position. Since the particle is moving, its position changes, and thus the direction of the velocity vector continuously changes. Therefore, the instantaneous linear velocity vector is not constant.
• Linear Momentum Vector: Linear momentum ($\vec{p}$) is defined as the product of mass ($m$) and velocity ($\vec{v}$), i.e., $\vec{p} = m\vec{v}$. Since the mass ($m$) is constant and the velocity vector ($\vec{v}$) is changing its direction (as discussed above), the linear momentum vector is also not constant.

Examining Acceleration in Circular Motion

Even with uniform speed, circular motion requires acceleration to change the direction of velocity.

• Centripetal Acceleration Vector: A particle moving in a circle with uniform speed experiences centripetal acceleration ($\vec{a}_c$), which is directed towards the center of the circle. The magnitude of this acceleration is constant, given by $a_c = \frac{v^2}{r}$, where $v$ is the uniform speed and $r$ is the radius of the circle. However, like the velocity vector, the direction of the centripetal acceleration vector also continuously changes, always pointing towards the center from the particle's current position. Therefore, the centripetal acceleration vector is not constant.

Determining Constant Angular Momentum

Angular momentum is a crucial quantity in rotational motion.

• Angular Momentum Vector about the Center: The angular momentum ($\vec{L}$) of a particle about a point (in this case, the center of the circle) is defined as $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{r}$ is the position vector from the center to the particle, and $\vec{p}$ is the linear momentum ($m\vec{v}$).
  In circular motion about the center:
  • The position vector $\vec{r}$ has a constant magnitude (the radius $r$).
  • The linear momentum vector $\vec{p} = m\vec{v}$ has a constant magnitude ($mv$ because speed $v$ is uniform).
  • The position vector $\vec{r}$ and the velocity vector $\vec{v}$ (and thus $\vec{p}$) are always perpendicular to each other.
  • The motion occurs in a fixed plane. The vector $\vec{r}$ is always in this plane, and $\vec{v}$ is also always in this plane. The cross product $\vec{L} = \vec{r} \times \vec{p}$ results in a vector perpendicular to the plane of motion.

  Let's consider the magnitude of the angular momentum: $L = |\vec{r} \times \vec{p}| = |\vec{r}| |\vec{p}| \sin(\theta)$ Here, $r = |\vec{r}|$, $p = |\vec{p}| = mv$, and $\theta$ is the angle between $\vec{r}$ and $\vec{p}$. Since $\vec{r}$ and $\vec{v}$ are always perpendicular in circular motion, $\theta = 90^\circ$ or $\frac{\pi}{2}$ radians, so $\sin(\theta) = 1$. Thus, the magnitude is $L = r \cdot mv$. Since $r$ (radius), $m$ (mass), and $v$ (uniform speed) are all constant, the magnitude $L$ is constant.

  Now consider the direction. The angular momentum vector $\vec{L}$ is perpendicular to the plane containing $\vec{r}$ and $\vec{p}$. Since the particle moves in a fixed circle (a fixed plane), and $\vec{r}$ and $\vec{p}$ are always in this plane, the resulting vector $\vec{L}$ is always perpendicular to this plane. As the plane of motion does not change, the direction of $\vec{L}$ is constant.

  Since both the magnitude and direction of the angular momentum vector about the center are constant, the angular momentum vector itself is constant.

Conclusion on Constant Quantities

Based on the analysis, when a particle moves in a circle with uniform speed, its instantaneous linear velocity vector, centripetal acceleration vector, and linear momentum vector are constantly changing in direction. Only the angular momentum vector about the center of the circle remains constant.