Answer:
The value of x is [tex]x = 0.1955\ m[/tex]
The value of the velocity at the top is [tex]v_{top} = 2.25 \ m/s[/tex]
Explanation:
The diagram of this process is on the first uploaded image
From the question we are told that
The mass of the block is [tex]m_b = 0.490kg[/tex]
The distance of compression is [tex]x[/tex]
The force constant is [tex]k = 450 N/m[/tex]
The radius of the circular track is [tex]R =1.00m[/tex]
The speed at the bottom of the track is [tex]v_A = 13.4 \ m/s[/tex]
The frictional force experienced is [tex]F_f = 7.00 \ N[/tex]
Now looking at this process we see that the potential energy of the spring is been transformed into the kinetic energy of the block . So,
[tex]PE \ of \ spring = KE \ of \ block[/tex]
mathematically i.e
[tex]\frac{1}{2}k x^2 = \frac{1}{2} m_bv^2_A[/tex]
[tex]0.5 * 450 * x^2 = 0.5 * 0.490 * 13.4^2[/tex]
Making x the subject of the formula
[tex]x = \sqrt{\frac{0.5 * 0.490 * 13.4^2}{ 0.5 * 450} }[/tex]
[tex]= 0.1955\ m[/tex]
The average workdone by friction is [tex]W_f = F_f * \pi[/tex]
Here [tex]\pi is \ the\ net\ displacement[/tex]
[tex]W_f = 7 * 3.142[/tex]
[tex]= 21.99 J[/tex]
The kinetic energy at the bottom is
[tex]KE = \frac{1}{2} * 0.490 * 13.4^2[/tex]
[tex]= 43.99 \ N[/tex]
The potential energy gained at the top of the circle is
[tex]PE = m_bgh[/tex]
Here h is the height which is equal to d(diameter) = 2r = 2 × 1 = 2 m and
g is acceleration due to gravity [tex]= 9.8m/s^2[/tex]
Now substituting values
[tex]PE = 0.490 * 9.8 *2[/tex]
[tex]=9.604 J[/tex]
Since energy is can not be created nor destroyed but transformed according to first law of thermodynamics
[tex]KE_{at \bottom} = PE _ {\ at \ top } + W_f + KE_{\ at \ top}[/tex]
[tex]43.99 = 9.604 +21.99 + [\frac{1}{2} m_b * v_{top} ^2][/tex]
[tex]12.369= [2.45* v_{top} ^2][/tex]
[tex]v_{top} =\sqrt{\frac{12.369}{2.45} }[/tex]
[tex]= 2.25 \ m/s[/tex]
