Question

A $50kg$box hangs from a rope. What is the tension in the rope if:

a. The box is at rest?

b. The box moves up at a steady $5.0m/s$?

c. The box has $v_y=5.0m/s$and is speeding up at $5.0m/s^2$?

d. The box has $v_y=5.0m/s$ and is slowing down at $5.0m/s^2$ ?

Short answer

Rope's tension value when the box is,

(a) at rest,$T=490N$.

(b) moving with uniform velocity,$T=490N$.

(c) speeding up,$T=740N$.

(d) speeding down,$T=240N$.

Step-by-step solution

Tension :

The pulling force transferred axially by a string, cable, chain, or similar item, or by each end of a rod, truss member, or similar three-dimensional object is known as tension.

(a) Explanation :

Newton's first law of motion states that an item remains at rest or in uniform motion until it is acted with by a net external force.

The quantity of force exerted on an object is equal to the product of its mass and the acceleration generated, according to Newton's second equation of motion.

$F=ma$

In this scenario, the item is a box that is suspended vertically by a rope. The free body diagram of the box is shown below.

$T=mg+m{a}_y$

Mass of box $\left(m\right)=50kg$,

Acceleration due to gravity $\left(g\right)=9.8m/s^2$,

rest velocity $v_y=0m/s$,

acceleration $a_y=0m/s^2$.

$T=m\left(g+a_y\right)$

$$\begin{gathered} T=50\left(9.8+0\right) \\ T=490N \end{gathered}$$

Hence, rope's tension value when the box is at rest is$490N$.

(b) Explanation : 

Mass of box $\left(m\right)=50kg$,

Acceleration due to gravity $\left(g\right)=9.8m/s^2$,

steady velocity $v_y=5m/s$,

acceleration $a_y=0m/s^2$.

Substituting values in equation,

$T=m\left(g+a_y\right)$

$$\begin{gathered} T=50\left(9.8+0\right) \\ T=490N \end{gathered}$$

Hence, rope's tension value when the box is moving with uniform velocity is$490N$.

(c) Explanation : 

Mass of box $\left(m\right)=50kg$,

Acceleration due to gravity $\left(g\right)=9.8m/s^2$,

velocity $v_y=5m/s$,

acceleration $a_y=5m/s^2$.

Substituting values in equation,

$T=m\left(g+a_y\right)$

$$\begin{gathered} T=50\left(9.8+5\right) \\ T=740N \end{gathered}$$

Hence, rope's tension value when the box is speeding up is$740N$.

(d) Explanation :

Mass of box $\left(m\right)=50kg$,

Acceleration due to gravity $\left(g\right)=9.8m/s^2$,

velocity $v_y=5m/s$,

acceleration $a_y=-5m/s^2$.

Substituting values in equation,

$T=m\left(g+a_y\right)$

$$\begin{gathered} T=50\left(9.8-5\right) \\ T=240N \end{gathered}$$

Hence, rope's tension value when the box is speeding down is$240N$.