Soru
![A solid sphere with radius r=2m rolls smoothly from rest from a ramp at height h=10m that is the vertical distance between the COM
of the sphere and the bottom of the ramp . When the sphere reached the bottom of the ramp, the sphere introduces a circular path
with R. When the sphere reaches the the top of the circular path with the speed of 3m/s determine the radius of the circle (R)
sphere.COM=(2/5)mr^2. assume that the sphere only makes smooth rolling]
Select one:
a. 4.8 m
b. 4.1 m
c. 5.7 m
d. 4.7 m
e. 3.5 m](https://static.questionai.com.tr/resource%2Fqaiseoimg%2F202412%2Fsolid-sphere-radius-r2m-rolls-smoothly-rest-ramp-height-tI7k5sTYMi0l.jpg?x-oss-process=image/resize,w_600,h_600/quality,q_50/format,webp)
A solid sphere with radius r=2m rolls smoothly from rest from a ramp at height h=10m that is the vertical distance between the COM of the sphere and the bottom of the ramp . When the sphere reached the bottom of the ramp, the sphere introduces a circular path with R. When the sphere reaches the the top of the circular path with the speed of 3m/s determine the radius of the circle (R) sphere.COM=(2/5)mr^2. assume that the sphere only makes smooth rolling] Select one: a. 4.8 m b. 4.1 m c. 5.7 m d. 4.7 m e. 3.5 m
Çözüm
4.4
(248 Oylar)
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Hediye
Elit · 8 yıl öğretmeni
Uzman doğrulaması
Cevap
To solve this problem, we need to use the principles of conservation of energy and the relationship between the center of mass (COM) and the radius of the sphere.Given information:- Radius of the sphere,
- Height of the ramp,
- Speed of the sphere at the top of the circular path,
Step 1: Calculate the potential energy at the top of the ramp.Potential energy at the top of the ramp =
where
is the mass of the sphere and
is the acceleration due to gravity.Step 2: Calculate the kinetic energy at the bottom of the ramp.Since the sphere rolls smoothly, the kinetic energy at the bottom of the ramp is the sum of the translational kinetic energy and the rotational kinetic energy.Kinetic energy at the bottom of the ramp =
where
is the moment of inertia of the sphere and
is the angular velocity.Step 3: Use the conservation of energy principle to find the radius of the circular path.Potential energy at the top of the ramp = Kinetic energy at the bottom of the ramp
Step 4: Substitute the values and solve for the radius of the circular path.Given that the COM of the sphere is
, we can use this relationship to find the moment of inertia of the sphere.
Substituting the values, we get:
Solving for the radius of the circular path, we get:
Therefore, the correct answer is:b. 4.1 m