Consider a circular hoop, a disk, and a sphere all having the same mass and radius. These objects all start at rest at the top of a ramp and

Question

Consider a circular hoop, a disk, and a sphere all having the same mass and radius. These objects all start at rest at the top of a ramp and are released to roll down the ramp at the same instant. If they all roll straight down the ramp without slipping or colliding, which one reaches the bottom first?

in progress 0
Farah 2 weeks 2021-08-28T18:33:10+00:00 1 Answers 0 views 0

Answers ( )

    0
    2021-08-28T18:34:59+00:00

    Answer:

    we see that the sphere is the body that arrives with the highest speed, therefore it must be the body that takes the least time,

    Explanation:

    Let’s analyze this exercise they ask us which body reaches the bottom of the ramp first, for which we must calculate the speed of the body in the final part, use the concepts of energy conservation

    starting point. Highest part of the ramp

          E_{m1} = ​​U = m g h

    at this point there is no kinematic energy because the body starts from rest

    Final point. Lower part of the ramp

          E_{m2} = K = ½ m v² + ½ I w²

    we fix the reference system at this point, as the body is rolling, it has energy of connection and rotation

    energy is conserved

               E_{m1} = E_{m2}²

              mgh = ½ m v² + ½ I w²

    the angular and linear variables are related

              v = wr

              w = v / r

             mg h = ½ m v² + ½ I v² / r²

              mg h = ½ m v² (1 + I / m r²)

               v² = 2gh / (1 + I / mr²)

    Since all bodies start from rest, the body that has more speed at the end of the ramp, is the body that takes less time.

    To find the speed we look in the tables for the moment of inertia of the bodies

    circular loop    I = mr²

    disk                   I = ½ m r²

    sphere              I = 2/5 mr²

    let’s calculate the speed for each body

    tie

          v² = 2g h (1+ mr² / mr²)

           v = √ (2gh) / √2

           v = 0.707 ra (2gh

    disk

          v² = 2gh / (1 + ½ mr² / mr²)

          v = ra 2gh / ra 3/2

          v = 0.816 ra 2gh

    sphere

          v² = 2gh / (1 + 2/5 mr₂ / mr)

          v = ra 2gh / ra 7/5

          v = 0.845 ra 2gh

    we see that the sphere is the body that arrives with the highest speed, therefore it must be the body that takes the least time, since

         t = d / v

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )