Question

Consider a ring, sphere and Solidey clinder all with the same mass. They are all held at the top of the inclined Plane which is at 20° to the horizontal. the top of the inclined Plane is Im high. The shapes are released simultaneously and allowed to roll down the inclined plane. Assume the abjects roll with out slipping and that they are all made from the same material. Assume the coefficient of static friction bin the objects and the plane is 0-3-

a) worklout what order

they would get to the bottom of the slope.

b) How long will it take each shape to reach the bottom of the Slope ?

c) which shapes have the greater moment of inertia ?

d) determine the linear acceleration(a)
e) calculate the tangential (linear) Veloci ty of each shapes-​

Answer 1

c

Step-by-step explanation:

Answer 2

a) The order in which the shapes reach the bottom of the slope will be the sphere, solid cylinder, and ring.

b) The time it takes for each shape to reach the bottom of the slope can be calculated using the following equation:

t = (2d / g)^(1/2)

Where t is the time, d is the height of the inclined plane (1m in this case), and g is the acceleration due to gravity (9.8 m/s^2).

For the sphere:

t = (2 x 1 / 9.8)^(1/2) = 0.45 seconds

For the solid cylinder:

t = (2 x 1 / 9.8)^(1/2) x (5/7) = 0.36 seconds

For the ring:

t = (2 x 1 / 9.8)^(1/2) x (2/5) = 0.28 seconds

c) The moment of inertia depends on the shape of the object and how the mass is distributed around its axis of rotation. For a solid sphere, the moment of inertia is given by I = (2/5)MR^2, for a solid cylinder it is I = (1/2)MR^2, and for a ring it is I = MR^2. Therefore, the order of increasing moment of inertia is the ring, the solid cylinder, and the sphere.

d) The linear acceleration of each shape can be calculated using the following equation:

a = gsinθ / (1 + I / MR^2)

Where a is the linear acceleration, g is the acceleration due to gravity (9.8 m/s^2), θ is the angle of the inclined plane (20° in this case), I is the moment of inertia, M is the mass, and R is the radius.

For the sphere:

a = (9.8 x sin20) / (1 + (2/5)) = 2.34 m/s^2

For the solid cylinder:

a = (9.8 x sin20) / (1 + (1/2)) = 3.29 m/s^2

For the ring:

a = (9.8 x sin20) / (1 + 1) = 4.16 m/s^2

e) The tangential (linear) velocity of each shape at the bottom of the slope can be calculated using the following equation:

v = ωR

Where v is the tangential velocity, ω is the angular velocity, and R is the radius.

The angular velocity can be calculated using the following equation:

ω = (2a / R)^(1/2)

For the sphere:

ω = (2 x 2.34 / 0.05)^(1/2) = 21.8 rad/s

v = 21.8 x 0.05 = 1.09 m/s

For the solid cylinder:

ω = (2 x 3.29 / 0.05)^(1/2) = 30.7 rad/s

v = 30.7 x 0.05 = 1.53 m/s

For the ring:

ω = (2 x 4.16 / 0.05)^(1/2) = 36.4 rad/s

v = 36.4 x 0.05 = 1.82 m/s

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