A solid sphere at rest at the top 12.0 m inclined plane rolls down the incline. How high vertically does it rise of the other incline?
Huh?
The question is worded incorrectly.
So to further explain: there are two inclines, the bottom ends of both inclines are touching, so that as the ball rolls to the bottom of the first incline, it immediately begins to rise up the second incline. You are supposed to assume that the ball will rotate due to friction.
To determine how high the solid sphere rises vertically on the other incline, we need to consider the conservation of energy.
First, we need to calculate the initial potential energy (PE) of the sphere when it is at the top of the 12.0 m inclined plane. The potential energy can be calculated using the formula:
PE = mgh
Where:
m = mass of the sphere
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height (12.0 m)
Next, as the sphere rolls down the incline, it will convert some of its potential energy into kinetic energy (KE). The kinetic energy can be calculated using the formula:
KE = (1/2)mv^2
Where:
m = mass of the sphere
v = velocity
Since the sphere is rolling without slipping, the relationship between the linear velocity (v) and the angular velocity (ω) is given by:
v = ωr
Where:
r = radius of the sphere
Now, as the sphere reaches the other incline, it starts to climb. At the highest point of the climb, all of the initial kinetic energy will be converted back into potential energy.
To calculate the height (h') the sphere reaches vertically on the other incline, we equate the initial potential energy (PE) to the final potential energy:
PE = mgh'
Since the initial potential energy is converted to kinetic energy as the sphere rolls down and then back into potential energy as it climbs, we can equate the initial potential energy to the final potential energy:
mgh = (1/2)mv^2 + mgh'
By canceling out the mass (m) and rearranging the equation, we can solve for the height (h'):
h' = (v^2 / (2g)) + h
To determine the height (h') the sphere reaches vertically on the other incline, we need to calculate the velocity (v) at the bottom of the incline.
To find the velocity (v), we can use the conservation of energy principle again. The initial potential energy (PE) at the top of the 12.0 m inclined plane is equal to the sum of the final kinetic energy (KE) and the final potential energy at the bottom of the incline:
PE = KE + mgh
By rearranging the equation and canceling out the mass (m), we can solve for the velocity (v):
v = √(2gh)
Now we can substitute the value of v into the equation for determining the height (h'):
h' = (v^2 / (2g)) + h
By plugging in the given values and calculating the equation, we can find the height (h') the solid sphere rises vertically on the other incline.