PHYSICS - CLUTCH NON-CALC CH 12: ROTATIONAL INERTIA &...

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PHYSICS - CLUTCH NON-CALC

CH 12: ROTATIONAL INERTIA & ENERGY

EXAMPLE: TWO BLOCKS ON A PULLEY / ATWOOD’S MACHINE

EXAMPLE: Two blocks are connected by a light string, which is ran around a pulley, as shown below. The blocks have

masses m1 = 3 kg and m2 = 5 kg. The pulley is a solid cylinder (mass 4 kg and radius 8 m), that is free to rotate about a

fixed, perpendicular axis through its center. The system is released from rest, with m2 5 m above the ground. Calculate:

(a) the speed of m2 just before it hits the ground;

(b) the pulley’s speed just before m2 hits the ground.

m2

m1



PRACTICE: TWO BLOCKS ON A PULLEY / ROUGH TABLE

PRACTICE: Two blocks are connected by a light rope, which passes through a pulley, as shown. The pulley is a solid

cylinder 10 kg in mass and 2 m in radius. The 4 kg block is on a horizontal surface, and the surface-block coefficient of

friction is 0.5. The system is released from rest, with the 6 kg block initially 8 m above the floor. Calculate the speed the 6 kg

block will have just before hitting the floor.

6 kg

4 kg



EXAMPLE: CONSERVATION OF ENERGY / SPEED OF YO-YO

EXAMPLE: When you release a simple 100-g yo-yo from rest, it falls and rolls, unwinding the light string around its

cylindrical shaft, which is 2 cm in radius. If the yo-yo can be modeled after a solid disc, calculate its: (a) linear speed after it

has dropped 50 cm; (b) angular speed after its dropped 50 cm.



PRACTICE: CONSERVATION OF ENERGY / SWINGING ROD

PRACTICE: A small 10-kg object is connected to the right end of a thin rod of length 4 m and mass 5 kg. The rod is free to

rotate about a fixed perpendicular axis on its left end, as shown below. The rod is initially held at rest, horizontally. When the

rod is released, it falls, rotating about its axis, similar to a pendulum. What is the speed at the rod’s center of mass when the

rod is vertical? BONUS: What is object’s speed when the rod is vertical?



CONSERVATION OF ENERGY / ROLLING MOTION

● Remember: If an object moves while rotating (Rolling Motion) on a surface without slipping v,CM = __________

- In order for an object to start rotating, or to rotate faster (___), there needs to be ___________________ (_____).

- The role of _________ in Rolling Motion is to “convert” some _____ into _____ ( ______ into ______ ).

- Unless otherwise stated, _________ does this without dissipating any energy W____ = _____

- To summarize: If α ≠ 0 there is ___________ but _____________.

If “without slipping” there is ___________.

EXAMPLE: A solid cylinder of mass M and radius R is released from rest from the top of an inclined plane of length L that

makes an angle of Θ with the horizontal. The cylinder rolls without slipping. Derive expressions for the linear and angular

speeds that the cylinder will have at the bottom of the plane.



PRACTICE: SPHERE GOING UP A HILL / MAX HEIGHT

PRACTICE: A solid sphere of mass M = 10 kg and radius R = 2 is rolling without slipping with speed V = 5 m/s on a flat

surface when it reaches the bottom of an inclined plane that makes an angle of Θ = 37o with the horizontal. The plane has

just enough friction to cause the sphere to roll without slipping while going up. What maximum height will the sphere attain?



EXAMPLE: ROLLING SPHERE / ROUGH AND SMOOTH HILLS

EXAMPLE: A solid sphere of mass M and radius R is initially at rest at the top of a rough hill of height H1.The sphere rolls

down the rough hill, then rides on a smooth horizontal surface, then goes up a long, smooth hill. The first hill (rough) has

enough friction to cause the sphere to roll without slipping. What maximum height H2, in terms of H1 and other variables, will

the sphere attain on the second hill (smooth)? BONUS: Why are the two heights different?



PRACTICE: SPHERE ON LOOP-THE-LOOP / SPEED AT BOTTOM

PRACTICE: You may remember that the lowest speed that an object may have at the top of a loop-the-loop of radius R, so

that it completes the loop without falling, is √ . Calculate the lowest speed that a solid sphere must have at the bottom of

a loop-the-loop, so that it reaches the top with enough speed to complete the loop. Assume the sphere rolls without slipping.



CONCEPT: PARALLEL AXIS THEOREM ● The moment of inertia is a really important quantity to know, because it acts like _____________ in rotational equations

- However, moment of inertia isn’t a fixed quantity like mass; it changes depending on the rotation

- Typically, moments are given on exams or in homework, but only what I’d call “typical” moments of inertia

- How do we solve a problem with a non-typical moment of inertia? ● For instance, we know how to find the moment of inertia of a disk rotating about its central axis: __________________

- What if this disk weren’t rotating about its central axis, but was rotating about its rim?

- If we aren’t given this new moment of inertia, we can find it using the PARALLEL AXIS THEOREM

- where d is the distance between the center-of-mass axis and the new axis, which must be PARALLEL EXAMPLE 1: A disk has a mass M and a radius R. What is its moment of inertia about an axis perpendicular to the surface of the disk, at the rim of the disk? What about a parallel axis half-way to the rim of the disk? EXAMPLE 2: The moment of inertia of a thin rod of length L and mass M about an axis perpendicular to the rod, at the edge of the rod, is (1/3)ML2. What is the moment of inertia of the rod about a parallel axis halfway from the edge to the center of the rod?

● The PARALLEL AXIS THEOREM will gives us the “non-typical” moments of inertia:

𝐼 = ________________



General Form: I = [ ] m R2

MOMENT OF INERTIA ● Remember: Motion problems do NOT depend on Mass, but Energy (eg. K = ½ m v2) and Force (ΣF = ma) problems do.

- Remember: Mass is the amount of resistance to LINEAR acceleration, which we call (linear) _______________:

● In ROTATION, the amount of resistance to acceleration depends on mass AND on ____________________________.

- This combination is called ________________________ (___), and it’s the rotational equivalent of ___________!

- You can think of it as (rotational) ______________.

● There are two types of objects:

- Point Masses ( ________ ) I = __________, where r = ________________________________.

- Shapes/Rigid Bodies ( ________ ) I is found by Table Lookup

( R is radius )

EXAMPLE: A system is made of two small masses (MLEFT = 3 kg, MRIGHT = 4 kg) attached to

the ends of a 2-m long thin rod that is massless, as shown. Calculate the moment of inertia of

the system if it spins about a perpendicular axis through the center of the rod.



PRACTICE: MOMENT OF INERTIA / SIMPLE SYSTEM

PRACTICE: A system is made of two small masses (MLEFT = 3 kg, MRIGHT = 4 kg) attached to

the ends of a 5 kg, 2-m long thin rod, as shown. Calculate the moment of inertia of the system

if it spins about a perpendicular axis through the mass on the left.



EXAMPLE: MOMENT OF INERTIA / EARTH

EXAMPLE: The Earth has mass and radius 5.97 x 1024 kg and 6.37 x 106 m. The radial distance between the Earth and the

Sun is 1.50 x 1011 m. Calculate the Moment of Inertia of the Earth as it spins around:

(a) itself -- treat the Earth as a solid sphere (solid spheres have moment of inertia given by 2/5 mR2);

(b) the Sun -- treat the Earth as a point mass.



PRACTICE: MOMENT OF INERTIA / FIND MASS

PRACTICE: A solid disc 4 m in diameter has a moment of inertia equal to 30 kg m2 about an axis through the disc,

perpendicular to its face. The disc spins at a constant 120 RPM. Calculate the mass of the disc.



EXAMPLE: MOMENT OF INERTIA / WITH DENSITY

EXAMPLE: A planet is nearly spherical with nearly continuous mass distribution, with 8 x 107 m in radius and 10,000 kg/m3

in density. If the planet rotates around itself, calculate its moment of inertia around its central axis (Note: VSPHERE = 4/3 πR3).



CONCEPT: FINDING MOMENT OF INERTIA BY INTEGRATING ● The moment of inertia of an object can be found by knowing the formula for a particular situation - However, what if you don’t have a formula for a particular situation? Plus, where do those formulas come from? - You can find the moment of inertia for any object about any axis by using integration

● For a point mass, 𝑑𝑚, at a distance 𝑟 from the rotation axis, the moment of inertia is 𝑑𝐼 = 𝑟2𝑑𝑚

- A solid object is made up of an infinite number of these infinitesimal masses, 𝑑𝑚, each at a different 𝑟 - To find the total moment of inertia, we need to sum all of the 𝑑𝐼’s, or integrate

EXAMPLE 1: What is the moment of inertia for a ring of mass m and radius R, rotating about an axis through its center, perpendicular to the surface of the ring? The mass is uniformly distributed throughout the ring.

● This integral isn’t USUALLY as simple as pulling 𝑟2 out of the integral and saying ∫ 𝑑𝑚 = 𝑚 - Since mass is distributed across all radii, we need to find a way to relate 𝑑𝑚 to 𝑑𝑟 EXAMPLE 2: What is the moment of inertia of a disk of mass m and radius R, rotating about an axis through its center, perpendicular to the surface of the ring? The mass is uniformly distributed throughout the disk.

● The MOMENT OF INERTIA of some object, about some axis, is

𝐼 = ___________________ where 𝑟 will TYPICALLY change with 𝑚



EXAMPLE: MOMENT OF INERTIA OF A NON-UNIFORM DISK What is the moment of inertia of a NON-UNIFORM disk, of mass m and radius R, about an axis through its center,

perpendicular to the surface of the disk. The mass distribution is given by 𝜎 = 𝛼𝑟2. Give your answer entirely in terms of m and R.



EXAMPLE: MOMENT OF INERTIA / DISC WITH MASSES

EXAMPLE: The solid disc below has radius 4 m and mass 10 kg. Two small objects are placed on top of it. The object on

the left has mass 2 kg and is placed half way between the disc’s center and its edge. The other object has mass 3 kg and is

placed at the edge of the disc. Calculate the system’s (disc + masses) moment of inertia around the disc’s central axis .



PRACTICE: MOMENT OF INERTIA / HOOP WITH RODS

PRACTICE: You build a wheel out of a thin circular hoop of mass 5 kg and radius 3 m, and two thin rods of mass 2 kg and 6

m in length, as shown below. Calculate the system’s moment of inertia about a central axis, perpendicular to the hoop .



PRACTICE: MOMENT OF INERTIA / COMPOSITE DISC

PRACTICE: A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc

(solid) has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3 m.

Calculate the moment of inertia of this composite disc about a central axis perpendicular to the discs .



PRACTICE: MOMENT OF INERTIA / EQUILATERAL TRIANGLE

PRACTICE: Three small objects, all of mass 1 kg, are arranged as an equilateral triangle of sides 3 m in length, as shown.

The left-most object is on (0m, 0m). Calculate the moment of inertia of the system if it spins about the (a) X axis; (b) Y axis.



MOMENT OF INERTIA DEPENDS ON MASS DISTRIBUTION

● The Moment of Inertia has to do with how mass is distributed on an object or system, relative to the axis of rotation.

EXAMPLE: A solid disc has small masses arranged on it in two different ways. In which of the following will the Moment of

Inertia of the system, about a central axis perpendicular to the disc, be greater?



PRACTICE: MOMENT OF INERTIA / MASS DISTRIBUTION

PRACTICE: The objects below all have the same mass and radius. Mass is distributed evenly in all objects. Rank the

objects according to the Moment of Inertia they each have about a central axis perpendicular to them, highest to lowest.



INTRO TO ROTATIONAL KINETIC ENERGY

● Remember: If you have linear speed (____), you have linear kinetic energy KL = ½ m v 2

- If you have rotational speed (____), you have rotational kinetic energy KR = _________

- If you are moving AND rotating, you have _____ AND _____ K = __________

- Remember: For Point Masses (R=0), I = _________. For Shapes/Rigid Bodies, we get I from a Table Look-up.

EXAMPLE: A basketball player spins a basketball around itself, on top of his finger, without the ball moving sideways. The

ball has mass 0.62 kg, diameter 24 cm, and spins at 15 rad/s. Calculate the ball’s linear, rotational, and total kinetic energy.



PRACTICE: ROTATIONAL ENERGY / ENERGY IN FLYWHEEL

PRACTICE: A flywheel is a rotating disc used to store energy. What is the maximum energy you can store on a flywheel

built as a solid disc with mass 8 x 104 kg and diameter 5.0 m, if it can spin at a max of 120 RPM?



EXAMPLE: ROTATIONAL ENERGY / RE-DESIGN FLYWHEEL

EXAMPLE: You are tasked with re-designing a solid disc flywheel to decrease its radius by half. How much mass must the

new flywheel have, relative to the original design, so that it can store the same amount of energy?



PRACTICE: ROTATIONAL ENERGY / FIND MASS

PRACTICE: When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 10,000 J in kinetic energy .

Calculate the sphere’s mass.



ROTATIONAL ENERGY / ROLLING ON A SURFACE VS. ON AIR

● Remember: If a wheel-like object moves on a surface while rotating around itself Rolling Motion v,CM = _______

- If an object is on air, its v,CM is NOT tied to w. This tie ONLY happens if it rolls on a surface (and without slipping).

EXAMPLE: A solid sphere of mass 2 kg and radius 0.3 m rolls without slipping on a horizontal surface with 10 m/s.

Calculate the sphere’s linear, rotational, and total kinetic energy.



PRACTICE: ROTATIONAL ENERGY / BASEBALL ON AIR

PRACTICE: A 150-g baseball, 3.85 cm in radius, leaves the pitcher’s hand with 30 m/s horizontal and 20 rad/s clockwise.

Calculate the ball’s linear, rotational, and total kinetic energy.



EXAMPLE: RATIO OF ENERGIES / CYLINDER ON SURFACE

EXAMPLE: A solid cylinder of mass M and radius R rolls without slipping on a horizontal surface with speed V. Calculate

the ratio of its rotational kinetic energy to its total kinetic energy.



PRACTICE: RATIO OF ENERGIES / SPHERE ON SURFACE

PRACTICE: A hollow sphere of mass M and radius R rolls without slipping on a horizontal surface with angular speed W.

Calculate the ratio of its linear kinetic energy to its total kinetic energy.



TYPES OF MOTION AND ENERGY

● A Point Mass (R=0) in a circular path has rotational speed (w).

- It also has a tangential speed (v,TAN), which is linear

- Does it mean that it has both LINEAR Kinetic Energy and ROTATIONAL Kinetic Energy? ___________

- We have ONE type of motion, so only ONE type of Energy v,TAN is just the linear equivalent of w.

EXAMPLE: For each of the following, indicate it whether has (i) linear kinetic energy, (ii) rotational kinetic energy.

(a) Box in a straight line (b) Disc spinning around itself (c) Earth around itself (d) Earth around Sun

(e) Moon spinning around the Earth (f) A roll of toilet paper rolling on the floor



KINETIC ENERGY OF A POINT MASS

● A Point Mass in a circular path has rotational speed (w) and a linear equivalent (v,TAN), BUT only ONE type of motion.

- So it only has ONE type of kinetic energy, BUT you can calculate it using EITHER equations (KL or KR).

- This is because the 2 equations are equivalent. What you can’t do is have BOTH – it would be “double counting”.

EXAMPLE: A small 2-kg object spins horizontally around a vertical axis at a rate of 3 rad/s, maintaining a constant distance

of 4 m to the axis. Calculate the object’s kinetic energy using: (a) KL; (b) KR.



PRACTICE: ROTATIONAL ENERGY / ENERGY OF EARTH

PRACTICE: The Earth has mass 5.97 x 1024 kg, radius 6.37 x 106 m. The Earth-Sun distance is 1.5 x 1011 m. Calculate the

Earth’s kinetic energy as it spins around itself. BONUS: Find the Earth’s kinetic energy as it goes around the Sun.



CONSERVATION OF ENERGY WITH ROTATION

● For problems between 2 points where speed/height/spring compression changes Conservation of Energy equation!

- The only difference is that now Kinetic Energy can be KL, KR, or ____________ ( _______________ ) !

- Most important thing: We will re-write ____ and ____ in terms of each other (so we reduce two variables into one).

EXAMPLE: A solid disc is free to rotate around a fixed, perpendicular axis through its center. The disc has mass M = 5 kg,

radius R = 6 m, and is initially at rest. A long, light cable is wrapped several times around the cylinder. You pull on the cable

with a constant 10 N, in such a way that the cable unwinds horizontally at the top of the disc unwind without slipping. Ignore

any frictional forces. Use Conservation of Energy to find the angular speed of the pulley after you’ve pulled the rope for 8 m.



EXAMPLE: WORK TO ACCELERATE CYLINDER

EXAMPLE: A solid cylinder of mass 10 kg and radius 2 m is mounted and free to rotate on a perpendicular axis through its

center. If the cylinder is initially at rest, how much work is needed to accelerate the cylinder to 120 RPM? Ignore any frict ion.



PRACTICE: WORK TO STOP SPHERE

PRACTICE: How much work is needed to stop a hollow sphere of mass 2 kg and radius 3 m that spins at 40 rad/s around

an axis through its center?



EXAMPLE: CONSERVATION OF ENERGY / SHAPES RACE DOWN

EXAMPLE: Three objects of equal mass and radius, but different shapes, are all released from rest, at the same time, from

the top of an inclined plane. They all roll without slipping. Which of the following shapes will reach the bottom first? Why?

(1) solid cylinder

(2) hollow cylinder

(3) solid sphere



EXAMPLE: CYLINDERS ON A HILL / SLIDDING VS. ROLLING

EXAMPLE: Two cylinders of equal mass and radius are released from rest from the top of two hills having the same height.

Cylinder A rolls down without slipping, and B slides down without rolling. Which will reach the bottom with greater speed?



PRACTICE: CONSERVATION OF ENERGY / SHAPES RACE UP

PRACTICE: Two solid cylinders of same mass and radius roll on a horizontal surface just before going up an inclined plane.

Cylinder A rolls without slipping, but cylinder B moves along a slippery path, so it moves without rotating at all times. At the

bottom of the incline, both have the same speed at their center of mass. Which will go higher on the inclined plane? (Why?)



TORQUE WITH MOTION EQUATIONS

● Similar to Force problems, some Torque problems will require both _______________ AND rotational motion equations.

- Remember: The variable that connects Newton’s 2nd Law & Motion Equations is ____________________ (____).

EXAMPLE: A solid sphere, 200 kg in mass and 6 m in diameter, spins about an axis through

its center with 180 RPM clockwise. How much torque is needed to stop it in just 10 s?

UAM EQUATIONS

(1) w = wo + α t

(2) w2 = wo2 + 2 α ΔΘ

(3) ΔΘ = wo t + ½ α t2

*(4) ΔΘ = ½ ( wo + w ) t



PRACTICE: TORQUE WITH MOTION / ROPE ON DISC

PRACTICE: A light, long rope is wrapped around a solid disc, in such a way that pulling the

rope causes the disc to spin about a fixed axis perpendicular to itself and through its center.

The disc has mass 40 kg, radius 2 m, and is initially at rest, and the rope unwinds without

slipping. You pull on the rope with a constant 200 N. Use the rotational version of Newton’s

Second Law to calculate how fast (in rad/s) the disc be spinning after you pull 50 m of rope.

UAM EQUATIONS

(1) w = wo + α t

(2) w2 = wo2 + 2 α ΔΘ

(3) ΔΘ = wo t + ½ α t2

*(4) ΔΘ = ½ ( wo + w ) t



PRACTICE: TORQUE WITH MOTION / TWO FORCES ON A SYSTEM

PRACTICE: A system is made of two small, 3 kg masses attached to the ends of a 5 kg, 4 m

long, thin rod, as shown. The system is free to rotate about an axis perpendicular to the rod

and through its center. Two forces, both of magnitude F and perpendicular to the rod, are

applied as shown below. What must the value of F be to the system from rest to 10 rad/s in

exactly 8 complete revolutions?



PRACTICE: TORQUE WITH MOTION / ROTATING DOORS

PRACTICE: Two rotating doors, each 6.0 m long, are fixed to the same central axis of

rotation, as shown (top view). When you push on one door with a constant 100 N,

directed perpendicular from the face of the door and 50 cm from its outer edge, the

rotating door system takes 8 s to complete a full revolution from rest. The doors can be

modeled as thin rectangles (moments of inertia for thin rectangles, around two

different axes, are shown for reference). Calculate the mass of the system.

I = 1/3 m A2

A = side that extends away from axis

I = 1/12 m (A2 + B2)

A and B = sides on face of rectangle

50 cm



EXAMPLE: TORQUE WITH MOTION / STOPPING WITH FRICTION

EXAMPLE: A flywheel is a rotating disc that is used to store energy. Suppose a flywheel has 8 x 104 kg in mass, 5.0 m in

diameter, and is setup vertically, as shown below, free to spin around a fixed, frictionless, perpendicular axis through its

center. To slow down the flywheel, you push a block against its outer rim. If the coefficients of friction between the block and

the flywheel’s rim are 0.6 and 0.8, how hard would you have to push against the flywheel so that it comes to a complete

stop, from 300 RPM, in just 30 s? You may assume that the wheel’s entire mass is concentrated on its outer rim.



PRACTICE: TORQUE WITH MOTION / STOPPING WITH FRICTION

PRACTICE: A 1,000 kg disc that has a 5 m outer radius is mounted on a vertical, inner

axle 80 kg in mass and 1 m in radius. A motor acts on the axle to speed up or slow down

the system. Suppose the motor stops functioning when the system is spinning at 70 rad/s.

To bring it to a complete stop, you apply a constant 200 N friction to the surface of the

axle. How many revolutions will the system take to stop?

(Note there are two objects (two I’s) and that the larger disc is NOT a solid cylinder)

A thick-walled cylinder

I = ½ m (R12 + R2

2)

R1

R2



ROTATIONAL DYNAMICS WITH TWO MOTIONS ● In problems where Torque causes Angular Acceleration, we use ____________.

- But in some problems, the system has both rotational AND linear motion.

- We write ____________ for each ___ AND ____________ for each ___.

- You’ll end up with ___ AND ___, so we’ll replace ___ with ___:

- The SIGNS for ____ & ____ as well as ____ & ____ must be consistent.

EXAMPLE: For each of the following, you want to solve for acceleration. (a) Determine which equations you’d start with.

(b) Sketch a diagram for each object showing forces and torques acting on it, with the proper signs (+ / –).

(i) Three blocks and two pulleys on a desk:

(ii) A simple yo-yo:

(iii) A cylinder rolling downhill:

m



EXAMPLE: ACCELERATION OF BLOCK ON A PULLEY

EXAMPLE: A block of mass m is attached to a long, light rope that is wrapped several times around a pulley, as shown

above. The pulley has mass M, radius R, can be modeled as a solid cylinder, and is free to rotate about a fixed, frictionless

axis perpendicular to itself and through its center. When the block is released from rest, it begins to fall, causing the pulley

to unwind without slipping. Derive an expression for the: (a) acceleration of the block; (b) angular acceleration of the pulley.



PRACTICE: TWO BLOCKS ON A PULLEY (ATWOOD’S MACHINE)

PRACTICE: Two blocks of masses m1 and m2 are both attached to a long, light rope that is wrapped several times around a

pulley, as shown below. The pulley has mass M and radius R, can be modeled as a solid cylinder, and is free to rotate

about a fixed, frictionless axis perpendicular to itself and through its center. When the block is released from rest, it begins

to fall, causing the pulley to unwind without slipping. Derive an expression for the angular acceleration of the pulley .

m2 m1



PRACTICE: ACCELERATION OF A YO-YO

PRACTICE: When you release a simple 100-g yo-yo from rest, it falls and rolls, unwinding the light string around its

cylindrical shaft, which is 2 cm in radius. If the yo-yo can be modeled after a solid disc, calculate its linear acceleration.



ROTATIONAL DYNAMICS OF ROLLING MOTION ● In some problems, a disc-like object accelerates around a FREE axis.

- In these, the object will have both rotational AND linear motion.

- So we use ___________ AND ___________ for the SAME object!

- The direction of positive in BOTH will follow the direction of _____.

● Remember: Rolling Motion has an extra equation: v,CM = ________.

- When disc-like objects roll freely, Torque comes from ______.

- If α ≠ 0 there is ___________.

- If “without slipping” there is ___________.

EXAMPLE: When a solid cylinder of mass M and radius R is released from rest, it rolls down without slipping along an

inclined plane that makes an angle Θ with the horizontal. Derive an expression for the angular acceleration of the cylinder.



PRACTICE: ROTATIONAL DYNAMICS / SPHERE GOING UP HILL

PRACTICE: A hollow sphere 10 kg in mass and 2 m in radius rolls without slipping along a horizontal surface with 20 m/s.

It then reaches an inclined plane that makes 37o with the horizontal, as shown. If it rolls up the incline without slipping, how

long will it take to reach its maximum height? (Hint: You will need to first calculate its acceleration)



PHYSICS - CLUTCH NON-CALC CH 12: ROTATIONAL INERTIA &...

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