When the ball is at the position as specified in the image above, these forces are applied to the basketball: Weight (downwards), Normal Force (perpendicular to ramp), and Force of friction (parellel to ramp). Out of these three forces, as weight and normal force is applied to the center of mass of the ball and the friction force is applied to the end of the ball, only the friction applies torque to the ball. Using these informations, we can solve for acceleration.
∑τ = Iα
τ = Ffr * r,
I = (2/3)mr^2
α = a/r
Ffr * r = ((2/3)mr^2) * (a/r)
Ffr = (2/3)ma
As there is no perpendicular acceleration of the ball, let's consider a as the parallel acceleration.
∑F = ma
mgsinθ - Ffr = ma
mgsinθ - (2/3)ma = ma
gsinθ - (2/3)a = a
gsinθ = (5/3)a
a = (3gsinθ)/5 = 3*9.8*sin(37˚)/5 = 3.5387 m / s^2
Afterwards, we can find the time travelled using kinematics.
v0 = 0m/s, ∆d = 3.52m, a = 3.5387 m / s^2
∆d = v0*t + (1/2)at^2
3.52 = (1/2)(3.5387)t^2
t = 1.410477472 s
Therefore, a basketball takes 1.410477472 seconds to roll without slipping down an incline.
Hi,
Thank you for your question.
When the ball is at the position as specified in the image above, these forces are applied to the basketball: Weight (downwards), Normal Force (perpendicular to ramp), and Force of friction (parellel to ramp). Out of these three forces, as weight and normal force is applied to the center of mass of the ball and the friction force is applied to the end of the ball, only the friction applies torque to the ball. Using these informations, we can solve for acceleration.
∑τ = Iα
τ = Ffr * r,
I = (2/3)mr^2
α = a/r
Ffr * r = ((2/3)mr^2) * (a/r)
Ffr = (2/3)ma
As there is no perpendicular acceleration of the ball, let's consider a as the parallel acceleration.
∑F = ma
mgsinθ - Ffr = ma
mgsinθ - (2/3)ma = ma
gsinθ - (2/3)a = a
gsinθ = (5/3)a
a = (3gsinθ)/5 = 3*9.8*sin(37˚)/5 = 3.5387 m / s^2
Afterwards, we can find the time travelled using kinematics.
v0 = 0m/s, ∆d = 3.52m, a = 3.5387 m / s^2
∆d = v0*t + (1/2)at^2
3.52 = (1/2)(3.5387)t^2
t = 1.410477472 s
Therefore, a basketball takes 1.410477472 seconds to roll without slipping down an incline.