Questions: What is the velocity of the mass at a time -t? Express this velocity in terms of R, ω, t, and the unit vectors î and ĵ.

Transcript text: What is the velocity of the mass at a time $-t$ ? Express this velocity in terms of $R, \omega, t$, and the unit vectors $\hat{i}$ and $\hat{j}$.

Solution

Solution Steps

Step 1: Understand the Problem

The problem involves a mass moving in a circular path with radius $ R $ and angular velocity $ \omega $. We need to find the velocity of the mass at a time $ -t $, which is a time $ t $ before the stopwatch starts.

Step 2: Determine the Position Vector at Time $-t$

The position vector of the mass at any time $ t $ is given by: \[ \vec{r}(t) = R \cos(\omega t) \hat{i} + R \sin(\omega t) \hat{j} \] At time $-t$, the position vector becomes: \[ \vec{r}(-t) = R \cos(-\omega t) \hat{i} + R \sin(-\omega t) \hat{j} \] Using the trigonometric identities $\cos(-\theta) = \cos(\theta)$ and $\sin(-\theta) = -\sin(\theta)$, we have: \[ \vec{r}(-t) = R \cos(\omega t) \hat{i} - R \sin(\omega t) \hat{j} \]

Step 3: Calculate the Velocity Vector at Time $-t$

The velocity vector is the derivative of the position vector with respect to time. Thus, we differentiate $\vec{r}(-t)$ with respect to $ t $: \[ \vec{v}(-t) = \frac{d}{dt}[R \cos(\omega t) \hat{i} - R \sin(\omega t) \hat{j}] \] Using the chain rule, we get: \[ \vec{v}(-t) = -R \omega \sin(\omega t) \hat{i} - R \omega \cos(\omega t) \hat{j} \]