Height is 7.05 m and not 7.05 mm
Answer:
9.603 m/s
Explanation:
We are dealing with rotation, so velocity of centre of mass is given by;
v_cm = Rω
Since we are working with a solid cylinder, moment of inertia of the cylinder is; I = ½mR²
Since it is rolled from the top to the bottom, at the top it will have potential energy(mgh) while at the bottom it will have kinetic energy (rotational plus translational kinetic energy).
Using conservation of energy, we have:
P.E = K.E_t + K.E_r
Formula for rotational and kinetic energy here are;
K.E_t = ½mv²
K.E_r = ½Iω²
mgh = ½mv² + ½Iω²
Since we want to find translational speed(v), let's get rid of ω.
Earlier, we saw that v_cm = Rω
Thus; ω = v/R
Also, we know that I = ½mR².
Thus;
mgh = ½mv² + ½(½mR²)(v/R)²
This gives;
mgh = ½mv² + ¼mv²
Divide through by m to get;
gh = v²(½ + ¼)
gh = ¾v²
Making v the subject gives;
v = √(4gh/3)
v = √((4 × 9.81 × 7.05)/3)
v = 9.603 m/s
