Question

A uniform stick has a mass of $4.42\mathrm{kg}$ and a length of $1.23\mathrm{m}$. It is initially lying flat at rest on a frictionless horizontal surface and is struck perpendicularly by a puck imparting a horizontal impulsive force of impulse $12.8\mathrm{N}\cdot \mathrm{s}$ at a distance of $46.4\mathrm{cm}$ from the center. Determine the subsequent motion of the stick.

Short answer

The stick will move with a linear velocity of 2.89 ms^-1 and rotate with an angular velocity of 2.04 rads^-1, perpendicular to the direction of the impulse.

Step-by-step solution

Calculate the Linear Velocity

First calculate the linear velocity (v) of the stick after the impulse. This can be found by using the equation for impulse $I=mv$, where m is the mass of the stick, I is the impulse and v is the velocity. Rearranging the equation, $v=I/m$, we have $v=12.8Ns/4.42kg=2.89m{s}^{-1}$.

Calculate the Angular Velocity

Next, calculate the angular velocity ($\omega$) of the stick. We use the equation for angular impulse, which in this case is $I=rFt$, where r is the distance from the center to the point of impulse, F is the force and t is the time. Given that impulse is the product of force and time, we know that Ft = I. Therefore, $I=rI$. The initiation of rotation can be found by using the equation for the moment of inertia for a rod rotated about its center, $I=(1/12)m{L}^2$, where L is the length of the stick. Hence we can express the angular velocity as: $\omega =rI/((1/12)m{L}^2)$. Substituting the given values, we get: $\omega =(0.464m\ast 12.8Ns)/((1/12)\ast 4.42kg\ast (1.23m{)}^2)=2.04rad{s}^{-}1$.

Determine the Subsequent Motion

The stick will move linearly with a velocity of 2.89 ms^-1 and rotate with an angular velocity of 2.04 rads^-1. It's important to note that it will move in a direction perpendicular to the impulse given by the puck.

Key concepts

Angular Velocity Calculation

Angular velocity is a measure of how fast an object rotates or spins around a certain axis. It's commonly represented by the symbol $\omega$ and calculated in radians per second (rad/s). When an object is struck off-center, as we saw in our exercise with the stick, it gains both linear and angular momentum. To calculate the angular velocity resulting from the impulse, you use the equation $\omega =rI/I_0$, where $r$ is the distance to the point of impulse, $I$ is the impulse, and $I_0$ is the moment of inertia. In the same manner as the linear velocity is affected by mass (inertia in linear motion), angular velocity is impacted by the distribution of an object's mass around the rotation axis, encapsulated in the moment of inertia term.

Understanding angular velocity is crucial in various real-life scenarios such as determining the speed of wheels turning on a vehicle, calculating rotational speeds in machinery, and even in analyzing the orbits of celestial bodies.

Linear Velocity

Linear velocity refers to the rate at which an object moves along a path in a particular direction. In physics, it's often denoted by $v$ and measured in meters per second (m/s). Linear velocity can be calculated using the impulse-momentum theorem, which states that the impulse applied on an object is equal to the change in momentum of that object. The impulse, denoted as $I$, is the product of the force applied and the time period over which it acts. Hence, for the stick in our exercise, finding the linear velocity after an off-center hit involves the simple formula of impulse divided by mass, $v=I/m$.

This concept is widely applied in fields ranging from traffic accident analysis to sports physics, and is essential for understanding how objects move in space in response to forces.

Moment of Inertia

The moment of inertia, symbolized by $I_0$, is a physical quantity expressing an object's tendency to resist angular acceleration. It is analogous to mass in linear motion and depends on both the mass of the object and the distribution of this mass relative to the axis of rotation. For regular shapes, the moment of inertia can be calculated using standard equations. For instance, in our exercise with the stick which can be considered a slender rod, the moment of inertia about an axis through its center is given by $I_0=(1/12)m{L}^2$ where $m$ is the mass and $L$ is the length of the rod.

Understanding the moment of inertia is important for analyzing the dynamics of rotational motion in mechanical engineering, structural designs, and even in athletics, for equipment designs like golf clubs or discuses.

Impulse-Momentum Theorem

The impulse-momentum theorem is a fundamental principle in physics that describes the relationship between impulse and the momentum of an object. The theorem states that the impulse imparted to an object equals the change in momentum of the object. The equation representing this theorem can be written as $I=\mathrm{\Delta }p$, where $I$ is the impulse, and $\mathrm{\Delta }p$ is the change in momentum. Momentum, in turn, equals mass times velocity, $p=mv$. The impulse is a vector quantity, as is momentum, so direction matters and indicates that the force applied will cause motion in the direction of the applied force.

For our stick hit by the puck, the impulse leads to both linear motion and rotation. The theorem aids in understanding various phenomena such as the recoiling of a gun when fired, the landing of an athlete from a jump, and the effects of collisions in automotive safety.