Question

Two objects of equal mass m are at rest at the top of a hill of height h. Object 1 is a circular hoop of radius r, and object 2 is a solid disc, also of radius r. The object are released from rest and roll without slipping.

A) Provide expressions for the LINEAR VELOCITY of each object once it reaches the bottom of the hill. Careful - you should provide two answers!

B) Considering your results from part A, which object reaches the bottom of the hill first? Provide a BRIEF justification in English for your answer.

C) For EACH object, provide an expression for what percentage of its total energy is rotational after it reaches the bottom of the hill. Careful - you should provide two answers!

Answer #1

**let m is the mass of each objec and h is the hight of
the hill.**

**A) for hoop,**

**Apply conservation of energy**

**final kinetic energy = initial potential
energy**

**(1/2)*m*v^2 + (1/2)*I*w^2 = m*g*h**

**(1/2)*m*v^2 + (1/2)*m*r^2*w^2 = m*g*h**

**(1/2)*m*v^2 + (1/2)*m*(r*w)^2 = m*g*h**

**(1/2)*m*v^2 + (1/2)*m*v^2 = m*g*h**

**m*v^2 = m*g*h**

**v_hoop = sqrt(g*h)**

**for disk,**

**Apply conservation of energy**

**final kinetic energy = initial potential
energy**

**(1/2)*m*v^2 + (1/2)*I*w^2 = m*g*h**

**(1/2)*m*v^2 + (1/2)*m*r^2*w^2 = m*g*h**

**(1/2)*m*v^2 + (1/2)*(1/2)m*(r*w)^2 = m*g*h**

**(1/2)*m*v^2 + (1/4)*m*v^2 = m*g*h**

**(3/4)m*v^2 = m*g*h**

**v_disk = sqrt(4*g*h/3)**

**B) disk recahes the bottom first.**

**because, v_disk > v_hoop**

**C) for hoop = KE_rotationa/KE_total**

**= (1/2)*m*v^2/(m*v^2)**

**= 50%**

**for disk = KE_rotationa/KE_total**

**= (1/4)*m*v^2/((3/4)m*v^2)**

**= 33.3%**

A hollow sphere (mass M, radius R) starts from rest at the top
of a hill of height H. It rolls down the hill without slipping.
Find an expression for the speed of the ball's center of mass once
it reaches the bottom of the hill.

The following four objects (each of mass m) roll without
slipping down a ramp of height h:
Object 1: solid cylinder of radius r
Object 2: solid cylinder of radius 2r
Object 3: hoop of radius r
Object 4: solid sphere of radius 2r
Rank these four objects on the basis of their rotational kinetic
energy at the bottom of the ramp.

Two objects roll down a hill: a hoop and a solid cylinder. The
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at the bottom of the hill and rank them according to their
speeds.
[Hint: When an object is rolling, the
angular speed and the velocity of the center of mass are related by
, where is the radius of the object.]
Veyi= 4.3 m/s...

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b) Calculate the rotational inertia (moment of inertia) for the
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How long did it take for the faster of the two objects to reach
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A
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1. When the sphere reaches the bottom of the ramp, what are
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2. When the sphere reaches the bottom of the ramp, what is its
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3. When the sphere reaches the bottom of the ramp, what is its...

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