$\text { What force is required so that a particle of mass } m \text { has the following position function? }$ $$\mathrm { r } ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$$
A) $\mathrm { F } ( t ) = 30 m t \mathbf { i } + 20 m \mathbf { j } + 42 m t \mathbf { k }$
B) $\mathrm { F } ( t ) = 30 m t ^ { 2 } \mathbf { i } + 20 m \mathbf { j } + 42 m t \mathbf { k }$
C) $\mathrm { F } ( t ) = 42 m t \mathbf { i } + 42 m \mathbf { j } + 20 m t \mathbf { k }$
D) $\mathrm { F } ( t ) = m t ^ { 2 } \mathbf { i } + 5 m t \mathbf { j } + 10 m t ^ { 2 } \mathbf { k }$
E) $\mathrm { F } ( t ) = 30 m t \mathbf { i } + 20 \mathrm {~m} \mathbf { j } + t \mathbf { k }$

Question

Quizplus · Accepted Answer

To find the force required, we first need to find the acceleration of the particle by differentiating the position function twice with respect to time. The position function is given by $\mathrm { r } ( t ) = 5 t ^ { 3 } \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 7 t ^ { 3 } \mathbf { k }$.First derivative (velocity):$$ \mathrm { v } ( t ) = \frac { d \mathrm { r } ( t ) } { d t } = 15 t ^ { 2 } \mathbf { i } + 20 t \mathbf { j } + 21 t ^ { 2 } \mathbf { k } $$Second derivative (acceleration):$$ \mathrm { a } ( t ) = \frac { d \mathrm { v } ( t ) } { d t } = 30 t \mathbf { i } + 20 \mathbf { j } + 42 t \mathbf { k } $$The force required is given by Newton's second law, $ \mathrm { F } = m \mathrm { a } $, so:$$ \mathrm { F } ( t ) = m ( 30 t \mathbf { i } + 20 \mathbf { j } + 42 t \mathbf { k } ) = 30 m t \mathbf { i } + 20 m \mathbf { j } + 42 m t \mathbf { k } $$

What force is required so that a particle of mass m has the following position function? \text { What force is required so that a particle of mass } m \text { has the following position function? } What force is required so that a particle of mass m has the following position function?

$\text { What force is required so that a particle of mass } m \text { has the following position function? }$