Q31P - University Physics with Modern Physics

Question

A 5.60 kg bucket of water is accelerated upward by a cord of negligible mass whose breaking strength is 75.0 N. If the bucket starts from rest, what is the minimum time required to raise the bucket a vertical distance of 12.0 m without breaking the cord?

Step-by-Step Solution

Answer

The minimum required time to raise the bucket a vertical distance of 12.0 m without breaking the cord is  1.56 s.

Step 1: Significance of breaking strength:

Whenever a wire/rod pulls by a specific tensile force, then there would be a tension force develops in the wire/rod. The maximum value of tension force cannot be more than the breaking strength of the wire/rod.

Step 2: Identification of the given data:

The given data can be listed below as,

 

The mass of a bucket of water is m=5.60 kg.

The breaking strength of a cord is T=75.0 N.

The initial velocity of the bucket is u=0 (Since starting from rest)

The value of vertical height is h=12.0.

Acceleration due to gravity is $9.8 \mathrm{m}/{\mathrm{s}}^2$.

Step 3: Draw the free-body diagram of the bucket:

The free-body diagram of the bucket is given below.

Here,   is the acceleration of the bucket, W is the weight of the bucket and g is the gravitational acceleration whose value is $9.81 m/s^2$.

Step 4: Determination the acceleration of the bucket:

According to Newton second law, the relation of the upward acceleration of the bucket is expressed as,

$$\begin{gathered} \sum _{ }{\mathrm{F}}_{\mathrm{net}}=\mathrm{ma} \\ \mathrm{T}-\mathrm{W}=\mathrm{ma} \\ \mathrm{T}-\mathrm{mg}=\mathrm{ma} \\ \mathrm{a}=\left(\frac{\mathrm{T}-\mathrm{mg}}{\mathrm{m}}\right) \end{gathered}$$ 

Here, $\sum _{ }{\mathrm{F}}_{\mathrm{net}}$ is the net force acting on the bucket in the vertical upward direction.

 

Substitute all the known values in the above equation.

$$\begin{gathered} \mathrm{a}=\left[\frac{\left(75.0 \mathrm{N}\right)-\left(5.60 \mathrm{kg}\right)\left(981\mathrm{m}/{\mathrm{s}}^2\right)\left({\displaystyle \frac{1\mathrm{N}}{1 \mathrm{kg}.\mathrm{m}/{\mathrm{s}}^2}}\right)}{\left(5.60 \mathrm{kg}\right)}\right] \\ =3.58 \mathrm{N}/\mathrm{kg} \\ =\left(3.58 \mathrm{N}/\mathrm{kg}\right)\times \left(\frac{1\mathrm{m}/{\mathrm{s}}^2}{1 \mathrm{N}/\mathrm{kg}}\right) \\ =3.58 \mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$ 

Step 5: Determination the minimum required time to raise the bucket a vertical distance of 12.0 m without breaking the cord

The relation of minimum required time to raise the bucket a vertical distance of 12.0 m without breaking the cord is expressed as,

$\mathrm{h}=\mathrm{ut}+\frac{1}{2}{\mathrm{gt}}^2$ 

  

Here, t is the minimum required time to raise the bucket a vertical distance of 12.0 m without breaking the cord.

 

Substitute all the known values in the above equation.

$$\begin{gathered} \left(12.0 m\right)=\left(0\right)t+\frac{1}{2}\left(9.81m/s^2\right)t^2 \\ \left(12.0 m\right)=\left(4.905 m/s^2\right)t^2 \\ {t}^2=\left[\frac{\left(12.0 m\right)}{\left(4.905 m/s^2\right)}\right] \\ t=1.56 s \end{gathered}$$ 

 

Thus, the minimum required time to raise the bucket a vertical distance of 12.0 m without breaking the cord is1.56 s.