Rotational and potential energy problem - Mastering physics

[Kamkazemoose]

Homework Statement

"In a lab experiment you let a uniform ball roll down a curved track. The ball starts from rest and rolls without slipping. While on the track, the ball descends a vertical distance h. The lower end of the track is horizontal and extends over the edge of the lab table; the ball leaves the track traveling horizontally. While free falling after leaving the track, the ball moves a horizontal distance x and a vertical distance y."

"Calculate x in terms of h and y, ignoring the work done by friction."

h - height of ramp ball rolls down
y- height of track above ground
x - horizontal distance ball travels while falling
g - acceleration due to gravity
m - mass of ball
r - radius of ball

Homework Equations

m*g*h = 1/2*m*v^2+1/2*I*(omega)^2 (potential energy equal to kinetic energy and rotational energy)
I= 2/5*m*r^2 (moment of inertia for a sphere)
(omega)=v/r
x = x_0+v_0*t+1/2*a*t^2

The Attempt at a Solution

I solved for the velocity of the ball at the bottom of the ramp, mass and radius cancels out and got gh=6/10*v^2, so v=sqrt(10/6*gh). Then I tried to find the time it would take for the ball to fall y and got y=1/2gt^2 as there was no initial vertical velocity. So, i got t=sqrt((2y)/g). I then multiplied the velocity by the time because there should be no horizontal acceleration, and I got x = sqrt(6/10*gh)*sqrt((2y)/g) and mastering physics said "Your answer either contains an incorrect numerical multiplier or is missing one." so, I'm not sure what I should do, where I went wrong or whatever, so if anyone has any help, it would be appreciated, thanks.

 

[asleight]

Homework Statement

"In a lab experiment you let a uniform ball roll down a curved track. The ball starts from rest and rolls without slipping. While on the track, the ball descends a vertical distance h. The lower end of the track is horizontal and extends over the edge of the lab table; the ball leaves the track traveling horizontally. While free falling after leaving the track, the ball moves a horizontal distance x and a vertical distance y."

"Calculate x in terms of h and y, ignoring the work done by friction."

h - height of ramp ball rolls down
y- height of track above ground
x - horizontal distance ball travels while falling
g - acceleration due to gravity
m - mass of ball
r - radius of ball

Homework Equations

m*g*h = 1/2*m*v^2+1/2*I*(omega)^2 (potential energy equal to kinetic energy and rotational energy)
I= 2/5*m*r^2 (moment of inertia for a sphere)
(omega)=v/r
x = x_0+v_0*t+1/2*a*t^2

The Attempt at a Solution

I solved for the velocity of the ball at the bottom of the ramp, mass and radius cancels out and got gh=6/10*v^2, so v=sqrt(10/6*gh). Then I tried to find the time it would take for the ball to fall y and got y=1/2gt^2 as there was no initial vertical velocity. So, i got t=sqrt((2y)/g). I then multiplied the velocity by the time because there should be no horizontal acceleration, and I got x = sqrt(6/10*gh)*sqrt((2y)/g) and mastering physics said "Your answer either contains an incorrect numerical multiplier or is missing one." so, I'm not sure what I should do, where I went wrong or whatever, so if anyone has any help, it would be appreciated, thanks.

$$2E = 2m\vec{g}\vec{h} = m\vec{v}^2 + I\vec{\omega}^2 \left(=\frac{I\vec{v}^2}{\vec{r}^2}\right) = m\vec{v}^2 + I\vec{\omega}^2 + 2mgy$$. Check this out. :)

 

[thomate1]

Your approach to solving this problem is correct. However, there is a small mistake in your calculation for the time it takes for the ball to fall y. The correct equation is y = 1/2 * g * t^2, which means t = sqrt(2y/g). So when you multiply the velocity by the time, you should get x = sqrt(10/6 * gh) * sqrt(2y/g). This should give you the correct answer. Make sure to double check your units and numerical values to ensure accuracy.