Answer:
a. v = 7.5 m/s
b. w = 50 rad/s
c. 46.667 rad
Explanation:
Using the equations of energy in the motion to determine the speed, angular speed and the angle
Ep = m * g * h , ⇒ h = 7m * sin 35
Ep = 1.5kg * 9.8m/s^2 * 7 m * sin 35
Ep= 59.02 J
Ek = ½ * m * v^2 , ⇒Ek = ½ *1.5 kg* v^2
Ew = ½ * I * ω^2 For a solid sphere I = 2/5 * m * r^2 ⇒ I = 2/5 * 1.5 * 0.15^2 = 0.0135
ω = v/0.15, ω^2 = v^2/0.0225
Ek = ½ * 0.0135 * v^2/0.0225
Ek = 0.3 * v^2
Total E = 0.75 * v^2 + 0.3 * v^2
E = 1.05 * v^2
59.02 J = 1.05 * v^2
v = √56.2 = 7.5 m/s
ω = 7.5 / 0.15 = 50 rad/s
C= 2 * π * 0.15 = 0.3 * π
θ =[ 7 /(0.3 * π) ] * (2 π)
θ= 46.667 rad
The answers to your questions are as follows
A) The linear speed of the center of the sphere when at the bottom : 7.5 m/s
B) The angular speed of the sphere about its center when at the bottom : 50 rad/s
C) The angle in radians at which the sphere rolls : 46.67 rads
To determine the above we will apply the energy equations
Potential energy ( Ep ) = mgh , where h = 7m * sin 35
Therefore: ( Ep ) = 1.5 * 9.8 * 7 * sin 35 --- ( 1 )
= 59.02 J
Kinetic energy ( Ek ) = ½ * m * v² ----- ( 2 )
= ½ * 1.5 * v²
= 0.75 v²
Also
Ew = ½ * I * ω² --- ( 3 )
For a solid sphere
I = 2/5 * m * r²
where : I = 2/5 * 1.5 * 0.15² = 0.0135
also ω = v /0.15, therefore ω² = v² / 0.15²
Equation ( 3 ) becomes
Ek = ½ * 0.0135 * v² / 0.0225
= 0.3 * v²
The Total energy = 0.75 v² + 0.3 * v²
= 1.05 * v²
Given that :
Potential energy = Total energy
59.02 J = 1.05 * v²
v² = 56.2
Therefore the Linear speed ( v ) = [tex]\sqrt{56.2}[/tex] = 7.5 m/s
also The angular speed ( ω ) = 7.5 / 0.15 = 50 rad/s
and The angle in radians ( C ) = 2*[tex]\pi[/tex]*0.15
= 0.3 * [tex]\pi[/tex]
Therefore : θ = ( 7 / (0.3 * [tex]\pi[/tex]) ) * ( 2 π )
θ = 46.67 rads
Learn more about energy equations : https://brainly.com/question/20658056
