A 2kg hallow sphere of a radius 6cm starts from rest and rolls without slipping down a 10 degree incline. if the length of the incline is 50cm then the velocity of the center of mass of the hallow sphere at the bottom of the incline is?
To find the velocity of the center of mass of the hollow sphere at the bottom of the incline, we can use the principle of conservation of mechanical energy.
Step 1: Calculate the change in potential energy.
The potential energy is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical height. In this case, the vertical height is the height of the incline.
Given:
m = 2 kg (mass of the hollow sphere)
g = 9.8 m/s^2 (acceleration due to gravity)
θ = 10 degrees (angle of the incline)
h = Length of the incline x sin(θ) = 50 cm x sin(10°) = 50 cm x 0.1745 = 8.725 cm = 0.08725 m (vertical height)
PE = mgh = 2 kg x 9.8 m/s^2 x 0.08725 m
Step 2: Calculate the change in kinetic energy.
The kinetic energy is given by the formula KE = 1/2Iω^2, where I is the moment of inertia and ω is the angular velocity. Since the sphere is rolling without slipping, we can use the equation ω = v/r, where v is the linear velocity and r is the radius of the sphere.
Given:
r = 6 cm = 0.06 m (radius)
To calculate the moment of inertia for a hollow sphere, we use the formula I = 2/3mr^2.
I = 2/3 x 2 kg x (0.06 m)^2
Step 3: Apply the conservation of mechanical energy.
According to the principle of conservation of mechanical energy, the change in potential energy is equal to the change in kinetic energy.
PE = KE
mgh = 1/2Iω^2
Plug in the values we calculated:
2 kg x 9.8 m/s^2 x 0.08725 m = 1/2 (2/3 x 2 kg x (0.06 m)^2) x (v/r)^2
Simplify the equation and solve for v:
v = sqrt(2gh/5) = sqrt(2 x 9.8 m/s^2 x 0.08725 m / 5)
Calculate the final velocity:
v ≈ sqrt(0.341767) ≈ 0.585 m/s
Therefore, the velocity of the center of mass of the hollow sphere at the bottom of the incline is approximately 0.585 m/s.
if it rolling, then the angular velocity is related to velocity as
w=v/r
the energy gravity gives up is
mg(.50*sin10)
The energy in the sphere is
1/2 I w^2 + 1/2 m v^2=
1/2 I v^2/r^2+1/2 m v^2
look up I (moment of inertia) for a hollow sphere, set KE = initial PE, and solve for v. Your teacher is too easy.