Question

A 64.0kg box hangs from a rope. What is the tension in the rope if: the box is at rest, The box moves up a steady 5.10m/s ? The box has vy = 4.70m/s and is speeding up at 5.20m/s2 ? The y axis points upward. The box has vy = 4.70m/s and is slowing down at 5.20m/s2 ?

Answer 1

Concepts and reason

The concept required to solve this problem is Newton’s second law.

First rearrange the Newton’s second law equation to solve for the expression of the tension force.

Finally calculate the Tension for each case by substituting the values.

Fundamentals

The Newton’s second law states that the net force on an object is the product of mass of the object and final acceleration of the object. The equation of the Newton’s second law is,

$\sum {\vec F = m\vec a}$

Here, $\sum {\vec F}$ is the net force on the object, $m$ is mass of the object, and $\vec a$ is the acceleration of the object.

The weight of a body is the force of gravitation of the earth that is,

$\vec w = m\vec g$

Here, $w$ is the weight of the body, $m$ is the mass, and $g$ is the acceleration due to gravity.

The sign convention used here is as follows:

Take all the forces in upward direction as positive and all the force in downward direction as negative.

Use the Newton’s second law equation.

$\sum {\vec F = m\vec a}$

Substitute $\vec T - \vec w$ for $\sum {\vec F}$ in the above equation $\sum {\vec F = m\vec a}$ .

$\vec T - \vec w = m\vec a$

Use the weight equation $\vec w = m\vec g$ and substitute $m\vec g$ for $\vec w$ in the above equation $\vec T - \vec w = m\vec a$ and rearrange to solve for $\vec T$ .

$\begin{array}{c}\\\vec T - m\vec g = m\vec a\\\\\vec T = m\vec g + m\vec a\\\\ = m\left( {\vec g + \vec a} \right)\\\end{array}$

……(1)

(a)

The acceleration of the box is zero when the box is at rest. Use the equation (1) to calculate the tension in the string.

$\vec T = m\left( {\vec g + \vec a} \right)$

Substitute $64{\rm{ kg}}$ for $m$ , $9.8{\rm{ m/}}{{\rm{s}}^2}$ for $\vec g$ , and $0{\rm{ m/}}{{\rm{s}}^2}$ for $\vec a$ in the above equation.

$\begin{array}{c}\\\vec T = \left( {64{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^2} + 0{\rm{ m/}}{{\rm{s}}^2}} \right)\\\\ = 627.2{\rm{ N}}\\\end{array}$

(b)

The acceleration of the box is zero when the box is rising at constant speed. Use the equation (1) to calculate the tension in the string.

$\vec T = m\left( {\vec g + \vec a} \right)$

Substitute $64{\rm{ kg}}$ for $m$ , $9.8{\rm{ m/}}{{\rm{s}}^2}$ for $\vec g$ , and $0{\rm{ m/}}{{\rm{s}}^2}$ for $\vec a$ in the above equation.

$\begin{array}{c}\\\vec T = \left( {64{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^2} + 0{\rm{ m/}}{{\rm{s}}^2}} \right)\\\\ = 627.2{\rm{ N}}\\\end{array}$

(c)

The acceleration of the box is upward so it is positive. Use the equation (1) to calculate the tension in the string.

$\vec T = m\left( {\vec g + \vec a} \right)$

Substitute $64{\rm{ kg}}$ for $m$ , $9.8{\rm{ m/}}{{\rm{s}}^2}$ for $\vec g$ , and ${\rm{5}}{\rm{.20 m/}}{{\rm{s}}^2}$ for $\vec a$ in the above equation $\begin{array}{c}\\\vec T = \left( {64{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^2} + 5.20{\rm{ m/}}{{\rm{s}}^2}} \right)\\\\ = \left( {64{\rm{ kg}}} \right)\left( {15.0{\rm{ m/}}{{\rm{s}}^2}} \right)\\\\ = 960{\rm{ N}}\\\end{array}$

(d)

The acceleration of the box is slowing it down so it is negative. Use the equation (1) to calculate the tension in the string and substitute $- \vec a$ for acceleration.

$\vec T = m\left( {\vec g - \vec a} \right)$

Substitute $64{\rm{ kg}}$ for $m$ , $9.8{\rm{ m/}}{{\rm{s}}^2}$ for $\vec g$ , and ${\rm{5}}{\rm{.20 m/}}{{\rm{s}}^2}$ for $\vec a$ in the above equation $\begin{array}{c}\\\vec T = \left( {64{\rm{ kg}}} \right)\left( {9.8{\rm{ m/}}{{\rm{s}}^2} - 5.20{\rm{ m/}}{{\rm{s}}^2}} \right)\\\\ = \left( {64{\rm{ kg}}} \right)\left( {4.6{\rm{ m/}}{{\rm{s}}^2}} \right)\\\\ = 294.4{\rm{ N}}\\\end{array}$

Ans: Part a

The tension in the rope is $627.7{\rm{ N}}$ .

Part b

The tension in the rope is $627.7{\rm{ N}}$ .

Part c

The tension in the rope is ${\rm{960 N}}$ .

Part d

The tension in the rope is $294.4{\rm{ N}}$ .