Max Height of Rolling Ball & Kinetic Energy Calculation

[BMW25]

two question important !

1) A hollow ball is rolling along a horizontal surface at 3.4 m/s when it encounters an upward incline. If it rolls without slipping up the incline, what maximum height will it reach?
h= ...meter

2) A solid 2.4 kg sphere is rolling at 6.6 m/s . Find (a) its translational kinetic energy and (b) its rotational kinetic energy.


pleeeeeeeeeeeeease help me with that as soon as possible

 

[mgb_phys]

1, What is it's kinetic energy - if it stops at the top of the incline all this goes into potential energy what is the height.

2, You need the equation for rotational ke of a solid sphere (it's in wiki or your textbook)
Then since it is rolling you can get the radius from considering how fast a point on the circumference is moving along the plane.

 

[ogb p]

1) To determine the maximum height the ball will reach, we can use the conservation of energy principle. The ball's initial kinetic energy will be converted into both potential energy and rotational kinetic energy as it rolls up the incline. Therefore, we can use the equation:
mgh = 1/2mv^2 + 1/2Iω^2
where m is the mass of the ball, g is the acceleration due to gravity, h is the maximum height, v is the velocity of the ball, I is the moment of inertia, and ω is the angular velocity.

First, we need to find the moment of inertia for a hollow sphere, which is given by I = 2/3mr^2, where r is the radius of the sphere. Substituting the given values, we get:
mgh = 1/2mv^2 + 1/2(2/3mr^2)(v/r)^2
Simplifying, we get:
h = v^2/2g + 2r/3

Plugging in the values of v = 3.4 m/s and g = 9.8 m/s^2, we get a maximum height of 1.5 meters.

2) To find the translational kinetic energy of the solid sphere, we use the equation:
KE = 1/2mv^2
where m is the mass of the sphere and v is its velocity. Plugging in the values of m = 2.4 kg and v = 6.6 m/s, we get a translational kinetic energy of 52.9 J.

To find the rotational kinetic energy, we use the equation:
KE = 1/2Iω^2
where I is the moment of inertia and ω is the angular velocity. For a solid sphere, the moment of inertia is given by I = 2/5mr^2. Substituting the values, we get:
KE = 1/2(2/5mr^2)(v/r)^2
Simplifying, we get:
KE = 2/5mv^2
Plugging in the values of m = 2.4 kg and v = 6.6 m/s, we get a rotational kinetic energy of 42.9 J.

Therefore, the total kinetic energy of the solid sphere is 95.8 J (52.9