A solid sphere of mass 0.604 kg rolls without slipping along a horizontal surface with a translational speed of 5.25 m/s. It comes to an incline that makes an angle of 38 with the horizontal surface. Neglecting energy losses due to friction,

(a) what is the total energy of the rolling sphere?

(b) to what vertical height above the horizontal surface does the sphere rise on the incline?

(a) KE= KE(transl) +KE(rot) = m•v²/2+Iω²/2=

= m•v²/2+ (2mR²)• v²/5•2•R²=
= m•v²/2+ mv²/5 =0.7 mv²
(b)
KE=PE
0.7•m•v²=mgh
h= 0.7•v²/g =…

To find the total energy of the rolling sphere and the vertical height it reaches on the incline, we can use the conservation of mechanical energy.

(a) The total mechanical energy of the rolling sphere consists of its kinetic energy (KE) and potential energy (PE). Since the sphere is rolling without slipping, it has both translational and rotational kinetic energy.

The translational kinetic energy (KET) of the sphere is given by the equation:
KET = (1/2) * mass * (velocity)^2

Substituting the given values:
KET = (1/2) * 0.604 kg * (5.25 m/s)^2
KET ≈ 8.120 J

The rotational kinetic energy (KER) of the sphere is related to its moment of inertia (I) and angular velocity (ω). For a solid sphere rolling without slipping, the equation is:
KER = (1/2) * I * ω^2

The moment of inertia for a solid sphere about its diameter is:
I = (2/5) * mass * (radius)^2

Since the sphere is rolling, its linear velocity and angular velocity are related by:
ω = v / r

Substituting the given values:
I = (2/5) * 0.604 kg * (r^2)
I ≈ 0.192 kg * (r^2)
ω = 5.25 m/s / r

Substituting in the equation for KER:
KER = (1/2) * (0.192 kg * (r^2)) * (5.25 m/s / r)^2
KER ≈ 0.511 J

Therefore, the total mechanical energy of the rolling sphere is the sum of its translational and rotational kinetic energy:
Total Energy (E) = KET + KER
E ≈ 8.120 J + 0.511 J
E ≈ 8.631 J

Hence, the total energy of the rolling sphere is approximately 8.631 J.

(b) To determine the vertical height that the sphere reaches on the incline, we can equate the initial mechanical energy to the potential energy at that height.

Initial energy (E_initial) = Total Energy (E) = 8.631 J

The potential energy (PE) at the top of the incline is given by:
PE = mass * g * height

Substituting the given angle and known values:
PE = 0.604 kg * 9.8 m/s^2 * height * sin(38°)

Using the energy conservation equation:
E_initial = PE
8.631 J = 0.604 kg * 9.8 m/s^2 * height * sin(38°)

Now, we can solve for the vertical height (height):
height = E_initial / (0.604 kg * 9.8 m/s^2 * sin(38°))
height ≈ 0.402 m

Therefore, the sphere rises vertically to approximately 0.402 m above the horizontal surface when rolling up the incline.