Question 1

A solid sphere of mass 2.5 kg and radius 0.12 m is at rest at the top of a ramp inclined 15°.It rolls to the bottom without slipping.The upper end of the ramp is 1.2 m higher than the lower end.Find the sphere's total kinetic energy when it reaches the bottom.

Accepted Answer

C

Question 2

A bus is designed to draw its power from a rotating flywheel that is brought up to its maximum speed (3000 rpm)by an electric motor.The flywheel is a solid cylinder of mass 250 kg and radius 0.500 m (I_cylinder = MR²/2).If the bus requires an average power of 10.0 kW,how long will the flywheel rotate?

Accepted Answer

A

Question 3

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N M applied to it.If it starts from rest,what kinetic energy will it have 8.8 s later?

Accepted Answer

B

Question 4

A meter stick is hinged at its lower end and allowed to fall from a vertical position.If its moment of inertia is ML²/3,with what angular speed does it hit the table?

Accepted Answer

The answer of A meter stick is hinged at its...

Question 5

A hoop with mass,M,and radius,R,rolls along a level surface without slipping with a linear speed,v.What is the ratio of rotational to linear kinetic energy? (For a hoop,I = MR². )

Accepted Answer

The rotational kinetic energy (K_rot) for a hoop is given by (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. For a hoop, I = MR^2. The linear kinetic energy (K_lin) is given by (1/2)Mv^2. Since the hoop rolls without slipping, v = ωR. Substituting ω = v/R into the rotational kinetic energy formula gives (1/2)MR^2(v/R)^2 = (1/2)Mv^2, which is equal to the linear kinetic energy. Therefore, the ratio of rotational to linear kinetic energy is 1:1.

Question 6

A rotating flywheel can be used as a method to store energy.If it is required that such a device be able to store up to a maximum of 2.00 × 10⁶ J when rotating at 400 rad/s,what moment of inertia is required?

Accepted Answer

The answer of A rotating flywheel can be used as...

Question 7

An initially installed flywheel can store 10⁶ J of kinetic energy when rotating at 300 rad/s.It is replaced by another flywheel of the same size but made of a stronger material.If its mass is the same as that of the original,and it is now capable of achieving a rotational speed of 395 rad/s,what maximum energy can be stored?

Accepted Answer

Kinetic energy is proportional to the square of angular velocity. So, the new maximum energy stored is:$$E = (395/300)^2 * 10^6 J = 1.73 * 10^6 J$$

Question 8

A solid sphere,a solid cylinder,and a hoop each have the same mass and radius.If they are spinning at the same angular velocity,which one has the greatest rotational kinetic energy?

Accepted Answer

The rotational kinetic energy of an object is given by the formula $KE_{rot} = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. The moment of inertia depends on the mass distribution relative to the axis of rotation. For objects with the same mass and radius, the hoop has the greatest moment of inertia ($I = mR^2$ for a hoop), making its rotational kinetic energy the highest among the given options.

Question 9

A hoop,solid sphere,and solid cylinder each have the same mass and same radius.When rolling down the same incline,which has the greatest torque from gravity on it?

Accepted Answer

The torque from gravity on an object rolling down an incline depends on the force of gravity and the radius of the object, not on the distribution of mass within the object. Since all three objects have the same mass and radius, they will experience the same torque from gravity.

Question 10

A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by 
When rolling down an incline.What is this object's moment of inertia?

Accepted Answer

The total kinetic energy of a rolling object is given by:

KE = (1/2)mv^2 + (1/2)Iω^2

where m is the mass of the object, v is its translational velocity, I is its moment of inertia about the axis of rotation, and ω is its angular velocity.

When the object is rolling without slipping, we have the relationship:

v = Rω

where R is the radius of the object. Substituting this into the expression for KE, we get:

KE = (1/2)mR^2ω^2 + (1/2)Iω^2

Simplifying, we obtain:

KE = (1/2)(mR^2 + I)ω^2

Comparing this with the given expression for KE, we see that:

(1/2)(mR^2 + I) = 11ea7a51_6367_38a6_89bd_7d77c89913cc_TB5259_11

Solving for I, we get:

I = 2KE - mR^2 = 11ea7a51_6367_38a8_89bd_4383a91afcc0_TB5259_11

Therefore, the answer is (B).

Question 11

The Earth's density increases with depth.If instead the Earth had uniform density,but the same angular velocity and mass,how would its rotational kinetic energy compare to the actual value?

Accepted Answer

The answer of The Earth's density increases with depth.If instead...

Question 12

A bowling ball has a mass of 7.0 kg,a moment of inertia of 2.8 × 10−² kg·m²,and a radius of 0.10 m.If it rolls down the lane without slipping at a linear speed of 2.5 m/s,what is its total kinetic energy?

Accepted Answer

The total kinetic energy is the sum of translational and rotational kinetic energy. Translational KE = 0.5 * m * v^2 = 0.5 * 7.0 kg * (2.5 m/s)^2 = 21.875 J. Rotational KE = 0.5 * I * ω^2, where ω = v/r = 2.5 m/s / 0.10 m = 25 rad/s. Rotational KE = 0.5 * 2.8 × 10^-2 kg·m^2 * (25 rad/s)^2 = 8.75 J. Total KE = 21.875 J + 8.75 J = 30.625 J, approximately 30.6 J.

Question 13

A solid sphere of mass 2.5 kg and radius 0.12 m starts from rest at the top of a ramp inclined 15° and rolls to the bottom.The upper end of the ramp is 1.6 m higher than the lower end.What is the linear speed of the sphere when it reaches the bottom of the ramp? (Note: I = 0.4MR² for a solid sphere,and g = 9.8 m/s². )

Accepted Answer

Explanation:The potential energy at the top of the ramp is:PE = mgh = (2.5 kg)(9.8 m/s^2)(1.6 m) = 39.2 JThe kinetic energy at the bottom of the ramp is:KE = 1/2mv^2The work done by gravity is equal to the change in kinetic energy:W = KE - PE = 1/2mv^2 - 39.2 JThe work done by gravity is also equal to:W = mgh = (2.5 kg)(9.8 m/s^2)(1.6 m) = 39.2 JTherefore,1/2mv^2 - 39.2 J = 39.2 JSolving for v, we get:v = sqrt(2(39.2 J) / 2.5 kg) = 4.7 m/s

Question 14

A cylinder with its mass concentrated toward the center has a moment of inertia of 0.2MR².If this cylinder is rolling without slipping along a level surface with a linear speed v,what is the ratio of its rotational kinetic energy to its linear kinetic energy?

Accepted Answer

The rotational kinetic energy ($K_{rot}$) of an object is given by $\frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. The linear kinetic energy ($K_{lin}$) is given by $\frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the linear velocity. For a cylinder rolling without slipping, $v = R\omega$, where $R$ is the radius. Given $I = 0.2mR^2$, substituting $R\omega = v$ into the rotational kinetic energy formula gives $K_{rot} = \frac{1}{2}(0.2mR^2)\left(\frac{v}{R}\right)^2 = \frac{1}{2} \cdot 0.2mv^2 = 0.1mv^2$. Therefore, the ratio of rotational to linear kinetic energy is $K_{rot}/K_{lin} = (0.1mv^2) / (0.5mv^2) = 1/5$.

Question 15

Three disks have the same mass and the same total kinetic energy.Disk #1 has radius R,disk #2 has radius 2R,and disk #3 has radius 3R.Which one has the highest v_{translational}?

Accepted Answer

The total kinetic energy for each disk is the sum of its translational kinetic energy and rotational kinetic energy. Since all disks have the same mass and total kinetic energy, and the only variable is the radius, the translational kinetic energy must be the same for all, as it is independent of the radius. This means all disks have the same translational velocity.

Question 16

A bucket of water with total mass 20.0 kg is attached to a rope,which in turn is wound around a 0.050-m radius cylinder,with crank,at the top of a well.The moment of inertia of the cylinder and crank is 0.21 kg M².The bucket is raised to the top of the well and released to fall back into the well.What is the kinetic energy of the cylinder and crank at the instant the bucket is moving with a speed of 7.0 m/s?

Accepted Answer

The potential energy of the bucket in the initial position is equal to the work done on it by the force of gravity as it falls back into the well. Thus, 
mgh = (1/2)mv^2 + (1/2)Iω^2
where m is the mass of the bucket and water, g is the acceleration due to gravity, h is the height of the well, v is the speed of the bucket at the bottom of the well, I is the moment of inertia of the cylinder and crank, and ω is the angular velocity of the cylinder and crank. 
Solving for the kinetic energy, we get 
(1/2)mv^2 + (1/2)Iω^2 = mgh 
(1/2)(20.0)(7.0)^2 + (1/2)(0.21)(7.0/0.050)^2 = (20.0)(9.81)(0.0) 
= 2 058 J.

Question 17

A rotating flywheel can be used as a method to store energy.If it has 2.0 × 10⁶ J of kinetic energy when rotating at 435 rad/s,and if a frictional torque of 4.0 N M acts on the system,in what interval of time would the flywheel come to rest?

Accepted Answer

The answer of A rotating flywheel can be used as...

Question 18

A hoop (I = MR²)is rolling along the ground at 10 m/s.It comes to a hill and starts going up.Assuming no losses to friction,how far does it travel vertically up the hill before it stops?

Accepted Answer

The answer of A hoop (I = MR²)is rolling along...

Question 19

A solid cylinder of mass 3.0 kg and radius 0.2 m starts from rest at the top of a ramp,inclined 15°,and rolls to the bottom without slipping.(For a cylinder I = 0.5MR². )The upper end of the ramp is 1.5 m higher than the lower end.Find the linear speed of the cylinder when it reaches the bottom of the ramp.(g = 9.8 m/s²)

Accepted Answer

We can use conservation of energy to solve this problem.

The initial potential energy of the cylinder is:

PEi = mgh = (3.0 kg)(9.8 m/s^2)(1.5 m) = 44.1 J

At the bottom of the ramp, all of the potential energy is converted to kinetic energy and rotational kinetic energy:

KEf + KEr = PEi

where KEf is the translational kinetic energy and KEr is the rotational kinetic energy.

The translational kinetic energy is given by:

KEf = (1/2)mv^2

where v is the linear speed of the cylinder.

The rotational kinetic energy is given by:

KEr = (1/2)Iω^2

where I is the moment of inertia of the cylinder about its axis (given by I = 0.5MR^2) and ω is the angular speed of the cylinder (related to the linear speed v by ω = v/R).

Substituting these expressions into the conservation of energy equation and solving for v, we get:

(1/2)mv^2 + (1/2)I(v/R)^2 = mgh

v^2 = (2/3)gh

v = √[(2/3)(9.8 m/s^2)(1.5 m)] ≈ 4.4 m/s

Therefore, the answer is B.

Question 20

A box slides down a frictionless incline,and a hoop rolls down another incline.Both inclines have the same height,and both the box and the hoop have the same mass.If both objects start from rest,upon reaching the bottom of the incline which one will have the greater kinetic energy and which one will have the greater speed?

Accepted Answer

The answer of A box slides down a frictionless incline,and...

Quiz 9: Solids and Fluids: Part A

A solid sphere of mass 2.5 kg and radius 0.12 m is at rest at the top of a ramp inclined 15°.It rolls to the bottom without slipping.The upper end of the ramp is 1.2 m higher than the lower end.Find the sphere's total kinetic energy when it reaches the bottom.

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N M applied to it.If it starts from rest,what kinetic energy will it have 8.8 s later?

A meter stick is hinged at its lower end and allowed to fall from a vertical position.If its moment of inertia is ML²/3,with what angular speed does it hit the table?

A hoop with mass,M,and radius,R,rolls along a level surface without slipping with a linear speed,v.What is the ratio of rotational to linear kinetic energy? (For a hoop,I = MR². )

A rotating flywheel can be used as a method to store energy.If it is required that such a device be able to store up to a maximum of 2.00 × 10⁶ J when rotating at 400 rad/s,what moment of inertia is required?

A solid sphere,a solid cylinder,and a hoop each have the same mass and radius.If they are spinning at the same angular velocity,which one has the greatest rotational kinetic energy?

A hoop,solid sphere,and solid cylinder each have the same mass and same radius.When rolling down the same incline,which has the greatest torque from gravity on it?

A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by When rolling down an incline.What is this object's moment of inertia?

The Earth's density increases with depth.If instead the Earth had uniform density,but the same angular velocity and mass,how would its rotational kinetic energy compare to the actual value?

A bowling ball has a mass of 7.0 kg,a moment of inertia of 2.8 × 10−² kg·m²,and a radius of 0.10 m.If it rolls down the lane without slipping at a linear speed of 2.5 m/s,what is its total kinetic energy?

A cylinder with its mass concentrated toward the center has a moment of inertia of 0.2MR².If this cylinder is rolling without slipping along a level surface with a linear speed v,what is the ratio of its rotational kinetic energy to its linear kinetic energy?

Three disks have the same mass and the same total kinetic energy.Disk #1 has radius R,disk #2 has radius 2R,and disk #3 has radius 3R.Which one has the highest v_{translational}?

A rotating flywheel can be used as a method to store energy.If it has 2.0 × 10⁶ J of kinetic energy when rotating at 435 rad/s,and if a frictional torque of 4.0 N M acts on the system,in what interval of time would the flywheel come to rest?

A hoop (I = MR²) is rolling along the ground at 10 m/s.It comes to a hill and starts going up.Assuming no losses to friction,how far does it travel vertically up the hill before it stops?

Quiz 9: Solids and Fluids: Part A

A solid sphere of mass 2.5 kg and radius 0.12 m is at rest at the top of a ramp inclined 15°.It rolls to the bottom without slipping.The upper end of the ramp is 1.2 m higher than the lower end.Find the sphere's total kinetic energy when it reaches the bottom.

A ventilation fan with a moment of inertia of 0.034 kg·m2 has a net torque of 0.11 N M applied to it.If it starts from rest,what kinetic energy will it have 8.8 s later?

A meter stick is hinged at its lower end and allowed to fall from a vertical position.If its moment of inertia is ML2/3,with what angular speed does it hit the table?

A hoop with mass,M,and radius,R,rolls along a level surface without slipping with a linear speed,v.What is the ratio of rotational to linear kinetic energy? (For a hoop,I = MR2. )

A rotating flywheel can be used as a method to store energy.If it is required that such a device be able to store up to a maximum of 2.00 × 106 J when rotating at 400 rad/s,what moment of inertia is required?

A solid sphere,a solid cylinder,and a hoop each have the same mass and radius.If they are spinning at the same angular velocity,which one has the greatest rotational kinetic energy?

A hoop,solid sphere,and solid cylinder each have the same mass and same radius.When rolling down the same incline,which has the greatest torque from gravity on it?

A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by When rolling down an incline.What is this object's moment of inertia?

The Earth's density increases with depth.If instead the Earth had uniform density,but the same angular velocity and mass,how would its rotational kinetic energy compare to the actual value?

A bowling ball has a mass of 7.0 kg,a moment of inertia of 2.8 × 10−2 kg·m2,and a radius of 0.10 m.If it rolls down the lane without slipping at a linear speed of 2.5 m/s,what is its total kinetic energy?

A cylinder with its mass concentrated toward the center has a moment of inertia of 0.2MR2.If this cylinder is rolling without slipping along a level surface with a linear speed v,what is the ratio of its rotational kinetic energy to its linear kinetic energy?

Three disks have the same mass and the same total kinetic energy.Disk #1 has radius R,disk #2 has radius 2R,and disk #3 has radius 3R.Which one has the highest vtranslational?

A rotating flywheel can be used as a method to store energy.If it has 2.0 × 106 J of kinetic energy when rotating at 435 rad/s,and if a frictional torque of 4.0 N M acts on the system,in what interval of time would the flywheel come to rest?

A hoop (I = MR2) is rolling along the ground at 10 m/s.It comes to a hill and starts going up.Assuming no losses to friction,how far does it travel vertically up the hill before it stops?

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N M applied to it.If it starts from rest,what kinetic energy will it have 8.8 s later?

A meter stick is hinged at its lower end and allowed to fall from a vertical position.If its moment of inertia is ML²/3,with what angular speed does it hit the table?

A hoop with mass,M,and radius,R,rolls along a level surface without slipping with a linear speed,v.What is the ratio of rotational to linear kinetic energy? (For a hoop,I = MR². )

A rotating flywheel can be used as a method to store energy.If it is required that such a device be able to store up to a maximum of 2.00 × 10⁶ J when rotating at 400 rad/s,what moment of inertia is required?

A bowling ball has a mass of 7.0 kg,a moment of inertia of 2.8 × 10−² kg·m²,and a radius of 0.10 m.If it rolls down the lane without slipping at a linear speed of 2.5 m/s,what is its total kinetic energy?

A cylinder with its mass concentrated toward the center has a moment of inertia of 0.2MR².If this cylinder is rolling without slipping along a level surface with a linear speed v,what is the ratio of its rotational kinetic energy to its linear kinetic energy?

Three disks have the same mass and the same total kinetic energy.Disk #1 has radius R,disk #2 has radius 2R,and disk #3 has radius 3R.Which one has the highest v_{translational}?

A rotating flywheel can be used as a method to store energy.If it has 2.0 × 10⁶ J of kinetic energy when rotating at 435 rad/s,and if a frictional torque of 4.0 N M acts on the system,in what interval of time would the flywheel come to rest?

A hoop (I = MR²) is rolling along the ground at 10 m/s.It comes to a hill and starts going up.Assuming no losses to friction,how far does it travel vertically up the hill before it stops?