Question

1. In an experiment where a smart cart moves freely down a ramp, why would it not be necessary to include friction in a calculation?

2. What would an equation for acceleration look like if you were to include friction?

3. The equation you used for acceleration was a=gsin(theta). Since the experiment was performed at 10 degrees or ess this falls under the small-angle approximation. This means that your acceleration can be approximated to be a=g(theta). Use the small-angle to find the linear approximation of the equation you found in question 2. compare this to a=g(theta) and use these equations to prove that friction will decrease the acceleration.  

small angle approximation: sin(theta)=(theta)

cos(theta)-1

Answer 1

1. it would not be necessary to include friction in a calculation because the value of friction force would be very less in comparison to the net downward force (mg sinθ) , so we can neglect friction and it will not affect our answer very much.

2. if we consider friction

then, in direction perpendicular to the inclination

N = mg cosθ

and, friction force (f) = μN = μmg cosθ

now , in direction along the inclination

net force (ma) = mg sinθ - f =  mg sinθ -  μ mg cosθ = m (g sinθ - μ g cosθ)

or, ma = m (g sinθ - μ g cosθ)

or, a = g sinθ - μ g cosθ = g( sinθ - μ cosθ)

this the required equation of acceleration.

3. now , sinθ = θ and cosθ = 1

putting into equation of acceleration we got,

a = g( sinθ - μ cosθ) = g (θ -  μ)

now in case when there is no friction

a' = g θ

and, in case when there is friction

a = g (θ -  μ)

hence, a' is greater than a

hence, friction will decrease the acceleration.