A solid sphere of uniform density starts from rest and rolls without slipping a distance of d = 2.7 m down a q = 23° incline. The sphere has a mass M = 4.8 kg and a radius R = 0.28 m. a) Of the total kinetic energy of the sphere, what fraction is translational?
KEtran/KEtotal =
5.03 NO
b) What is the translational kinetic energy of the sphere when it reaches the bottom of the incline?
KEtran = J
c) What is the translational speed of the sphere as it reaches the bottom of the ramp?
v = m/s
3.4 NO
Now let's change the problem a little.
d) Suppose now that there is no frictional force between the sphere and the incline. Now, what is the translational kinetic energy of the sphere at the bottom of the incline?
KEtran = J
a) 5/7
b) (5/7) m g h
c) sort(2E/m)
d) m g h
how do i find the hight for b?
H = 2.7 sin 23 m is the elevation change
To solve these questions, we'll need to use some principles of rotational and translational motion, as well as the concept of work and energy.
a) The total kinetic energy of the sphere can be divided into its translational kinetic energy and its rotational kinetic energy. The translational kinetic energy is given by the formula KEtran = (1/2) * M * V^2, where M is the mass of the sphere and V is its translational velocity. The rotational kinetic energy is given by the formula KErot = (1/2) * I * ω^2, where I is the moment of inertia of the sphere and ω is its angular velocity.
For a sphere rolling without slipping, the relationship between V and ω is V = R * ω, where R is the radius of the sphere.
To calculate the fraction of the total kinetic energy that is translational, we can use the formula:
KEtran/KEtotal = KEtran / (KEtran + KErot)
Substituting the expressions for KEtran and KErot, we get:
KEtran/KEtotal = (1/2) * M * V^2 / ((1/2) * M * V^2 + (1/2) * I * ω^2)
Using the relationship between V and ω, we can simplify this expression:
KEtran/KEtotal = (1/2) * M * (R * ω)^2 / ((1/2) * M * (R * ω)^2 + (1/2) * I * ω^2)
Substituting the known values for M, R, and I, we can calculate the numerical value for this fraction.
b) The translational kinetic energy of the sphere when it reaches the bottom of the incline can be calculated using the formula KEtran = (1/2) * M * V^2. We need to find the translational velocity V at the bottom of the incline, which can be calculated using the principles of motion.
c) The translational speed of the sphere as it reaches the bottom of the ramp can be found from the translational velocity V calculated in part b) by taking the square root of V^2.
d) If there is no frictional force between the sphere and the incline, the sphere will still have the same translational kinetic energy at the bottom of the incline. We can calculate this using the same formula as in part b).
I hope this explanation helps you understand the problem and how to solve it.