A 2.0 kg disc rolls without slipping on a horizontal surface, so that its center of mass has a constant linear speed of 6.0 m/s. What is the total kinetic energy of the disc?
i know that k= 0.5mv2 + 0.5 I w2
K=1/2 mv2 (1+(I/mr2)
but i am still getting he wrong answer because i convert I to miri2 so therefore it cancels with the denominator and k = 1/2mv2 but that is wrong. i am stuck
I for a solid disc is (1/2) m r^2
Try it with that.
If you use I = m r^2 (which would be correct for a hoop), you would get
total KE = mv^2, not(1/2) m v^2
To determine the total kinetic energy of the disc, you can use the formula K = 1/2 mv^2 + 1/2 Iω^2, where K is the total kinetic energy, m is the mass of the disc, v is the linear speed of the center of mass, I is the moment of inertia of the disc, and ω is the angular speed.
In this case, the disc is rolling without slipping, which means that its linear speed and angular speed are related. The relationship between linear speed and angular speed for a rolling object is ω = v / r, where r is the radius of the disc.
From the problem statement, you are given the linear speed of the center of mass as 6.0 m/s. To determine the angular speed, you need to know the radius of the disc. Let's assume the radius is given as r.
Using the relationship ω = v / r, you can substitute the given linear speed of the center of mass and the assumed radius into the equation. So, ω = 6.0 m/s / r.
Now, you need to determine the moment of inertia of the disc (I). The moment of inertia depends on the shape and mass distribution of the disc. For a solid disc, the moment of inertia is I = 1/2 mr^2, where m is the mass of the disc.
Substituting the given mass of the disc and the assumed radius into the equation, you get I = 1/2 * 2.0 kg * r^2.
Now, you have all the necessary values to calculate the total kinetic energy. Substituting the values into the formula K = 1/2 mv^2 + 1/2 Iω^2, you get K = 1/2 * 2.0 kg * (6.0 m/s)^2 + 1/2 * (1/2 * 2.0 kg * r^2) * (6.0 m/s / r)^2.
Simplifying the equation further, canceling terms and rearranging, you get:
K = 1/2 * 2.0 kg * (6.0 m/s)^2 + 1/8 * 2.0 kg * (6.0 m/s)^2.
Now, you can calculate this expression to find the total kinetic energy of the disc.