Question

A uniform solid sphere with a mass of M = 390 grams and a radius R = 19.0 cm is rolling without slipping on a horizontal surface at a constant speed of 4.00 m/s. It then encounters a ramp inclined at an angle of 15.0 degrees with the horizontal, and proceeds to roll without slipping up the ramp. Use g = 10.0 m/s2. How far does the sphere travel up the ramp (measure the distance traveled along the incline) before it stops for an instant?

Answer 1

l = 4.33 m

Explanation:

given,

mass of solid sphere = 390 gram = 0.39 kg

radius = R = 19 cm = 0.19 m

rolling with constant speed = 4 m/s

angle with horizontal = 15°

acceleration due to gravity = 10 m/s²

using energy conservation

`(1)/(2)I\omega^2 + (1)/(2)mv^2 = mgh`

I for sphere

`I = (2)/(5)mr^2` v = r ω

`(1)/(2)\ (2)/(5)mr^2* (v^2)/(r^2) + (1)/(2)mv^2 = mgh`

`(7)/(10)mv^2 = mgh`

`h = (0.7 v^2)/(g)`

`h = (0.7* 4^2)/(10)`

h = 1.12 m

`l=(h)/(sin 15^0)`

`l=(1.12)/(sin 15^0)`

l = 4.33 m

the sphere will travel 4.33 m on the ramp.