Physics

Problem

Consider a particle of constant mass

m

whose position vector at time

t

\vec{r}(t)

and velocity is

\vec{v}=\dfrac{d\vec{r}}{dt}

. A force

\vec{F}

acts on it, and its angular velocity about the origin is

\vec{\omega}

. a. (i) Define linear momentum

\vec{p}

and classify it, with justification, as a vector or a scalar. b. (ii) Show that the instantaneous power delivered to the particle is

P=\vec{F}\cdot\vec{v}

and hence establish that power is a scalar quantity. c. (iii) The torque acting on the particle about the origin is given by

\vec{\tau}=\vec{r}\times\vec{F}

. Prove that the quantity

\vec{\tau}\cdot\vec{\omega}

has no direction associated with it and therefore must be classified as a scalar. d. (iv) Summarise the classifications of

\vec{F}

\vec{v}

P

\vec{\tau}

and

\vec{\tau}\cdot\vec{\omega}

, stating one clear reason for each.