Respuesta :
Answer:
The time taken by the bar to reach the bottom t=4.886s
Given:
Displacement of the bar S=9.2m
Angle of inclination [tex]\theta=8.0^{\circ}[/tex]
Coefficient of friction factor [tex]\mu k=0.056[/tex]
To find:
How long it takes to reach the bottom ‘t’
Step by Step Explanation:
Solution:
We know that the formula for weight of the soap bar is given as
[tex]F_{g}=m g \sin \theta[/tex]
The frictional force acting on this soap bar is determined by
[tex]F_{f}=\mu m g \cos \theta[/tex]
To determine the constant acceleration of the bar, we derive as
[tex]F=m a[/tex]
Here [tex]F=F_{g}-F_{f}[/tex] and thus
[tex]F_{g}-F_{f}=m a[/tex][tex]m g \sin \theta-\mu m g \cos \theta=m a[/tex]
[tex]a=g \sin \theta-\mu g \cos \theta[/tex]
Where[tex]F_{g}[/tex]=Force imparted due to weight
[tex]F_{f}[/tex]=Frictional Force
m=Mass of the bar
g=Acceleration due to gravity
a=Acceleration of the bar
[tex]\sin \theta[/tex] and [tex]\cos \theta[/tex] are the angles involved in the system
If the bar starts from the rest
Equations of motion involved in calculating the displacement of the bar is given as
[tex]s=\frac{1}{2} a t^{2}[/tex], From this
[tex]a t^{2}=2 s[/tex]
[tex]t^{2}=\frac{2 s}{a}[/tex]
[tex]t=\sqrt{\frac{2 s}{a}}[/tex]
Where s= displacement or length moved by the bar
a=Acceleration of the bar
t=Time taken to reach bottom
Substitute all the known values in the above equation we get
[tex]t=\sqrt{\frac{2 \times 9.2}{a}}[/tex] and we know that
[tex]a=g \sin \theta-\mu g \cos \theta[/tex]
[tex]=9.8 \times \sin 8.0-0.056 \times 9.8 \times \cos 8.0[/tex]
[tex]=1.364-0.543[/tex]
[tex]a=0.821[/tex]
[tex]t=\sqrt{\frac{2 \times 9.2}{0.821}}[/tex]
[tex]t=\sqrt{\frac{19.6}{0.821}}[/tex]
[tex]t=\sqrt{23.87332}[/tex]
t=4.886s
Result:
Thus the time taken by the bar to reach the bottom is t=4.886s
