A solid sphere of uniform density starts from rest and rolls without slipping a distance of d = 4.8 m down a θ = 36° incline. The sphere has a mass M = 3.2 kg and a radius R = 0.28 m. If there is no frictional force, what is the translational kinetic energy of the sphere at the bottom of the incline?

Well, well, well, let's calculate that translational kinetic energy for our rolling sphere!

To find the translational kinetic energy, we need to consider that it rolls without slipping. So, we can relate the linear speed of the center of mass with the angular speed of the sphere.

First, let's find the linear velocity of the center of mass. We can do this by using the equation v = ωR, where ω is the angular speed and R is the radius.

The angular speed can be determined by relating it to the linear acceleration using ω = αt, where α is the angular acceleration and t is the time.

Since there is no friction, the angular acceleration will be zero. Why? Because if there's no friction, there's no torque, and no torque means no angular acceleration. Got it?

Now, let's find the time it takes for the sphere to roll a distance of d = 4.8 m down the incline. The time can be calculated using the equation d = 0.5at^2, where a is the acceleration and t is the time.

For the incline, we need to take into account the gravitational force. The acceleration can be found using the equation a = gsin(θ), where g is the acceleration due to gravity and θ is the angle of the incline.

Alright, putting it all together:

Acceleration a = gsin(θ) = 9.8 m/s^2 * sin(36°).

Time t can be obtained from the equation d = 0.5at^2, so t = sqrt(2d/a).

Now, let's get back to finding the linear velocity of the center of mass.

v = ωR. Here, ω = 0 because there's no angular acceleration (no friction, remember?). Thus, v = 0.

Wait, what?

Yup, without friction, the sphere won't have any translational kinetic energy at the bottom of the incline. It just rolls down calmly and comes to a stop.

And that, my friend, is the hilarious thing about no friction – it sucks all the fun out of our translational kinetic energy party!

To find the translational kinetic energy of the sphere at the bottom of the incline, we can use the principle of conservation of energy.

The initial potential energy of the sphere at the top of the incline is given by:

PE_initial = M * g * h

where M is the mass of the sphere, g is the acceleration due to gravity, and h is the height of the incline.

Since the sphere is rolling without slipping, the height of the incline can be calculated as:

h = d * sin(θ)

where d is the distance rolled down the incline and θ is the angle of the incline.

Plugging the values into the equation, we have:

h = 4.8 m * sin(36°)

Next, we find the final kinetic energy of the sphere at the bottom of the incline, which is the sum of its translational and rotational kinetic energies. Since there is no frictional force, all the initial potential energy is converted into kinetic energy at the bottom.

KE_final = PE_initial

Therefore, the translational kinetic energy of the sphere at the bottom is given by:

KE_translational = KE_final - KE_rotational

Now, let's find the rotational kinetic energy of the sphere. The sphere rolling without slipping, it has both translational and rotational motion. The rotational kinetic energy is given by:

KE_rotational = (1/2) * I * ω^2

where I is the moment of inertia and ω is the angular velocity.

For a solid sphere rolling without slipping, the moment of inertia is given by:

I = (2/5) * M * R^2

where R is the radius of the sphere.

To find the angular velocity, we can use the relationship between linear and angular velocity:

v = ω * R

Solving for ω:

ω = v / R

Since the sphere is rolling without slipping, its linear velocity at the bottom can be found using the equation:

v = √(2 * g * h)

Now, we have all the information needed to calculate the translational kinetic energy:

Step 1: Calculate the height of the incline:
h = 4.8 m * sin(36°)

Step 2: Calculate the linear velocity at the bottom:
v = √(2 * g * h)

Step 3: Calculate the angular velocity:
ω = v / R

Step 4: Calculate the moment of inertia:
I = (2/5) * M * R^2

Step 5: Calculate the rotational kinetic energy:
KE_rotational = (1/2) * I * ω^2

Step 6: Calculate the translational kinetic energy:
KE_translational = KE_final - KE_rotational

I hope this step-by-step solution helps you!

To calculate the translational kinetic energy of the sphere at the bottom of the incline, we need to find its linear velocity. We can find the linear velocity using the principle of conservation of energy.

First, let's calculate the potential energy of the sphere at the top of the incline. The potential energy can be determined using the equation:

PE = mgh,

where m is the mass of the sphere, g is the acceleration due to gravity, and h is the height of the incline. Since the sphere starts from rest, its initial kinetic energy is zero.

At the top of the incline, the height can be calculated using the formula:

h = d * sin(θ),

where d is the distance and θ is the angle of the incline. Substituting the values given:

h = 4.8 m * sin(36°) = 2.90 m.

Now we can calculate the potential energy:

PE = (3.2 kg) * (9.8 m/s^2) * (2.90 m) = 89.12 J.

Since there is no frictional force, all the potential energy will be converted to kinetic energy at the bottom of the incline. Therefore, the translational kinetic energy of the sphere at the bottom can be calculated as:

KE = PE = 89.12 J.

Hence, the translational kinetic energy of the sphere at the bottom of the incline is 89.12 J.

mgh = 7/10 mv^2

Use trig to find h
solve for v
find 1/2 mv^2