Question

You and your friend go sledding. Out of curiosity, you measure the constant angle θ that the snow-covered slope makes with the horizontal. Next, you use the following method to determine the coefficient of friction µk between the snow and the sled. You give the sled a quick push up so that it will slide up the slope away from you. You wait for it to slide back down, timing the motion. It turns out that the sled takes four times as long to slide down as it does to reach the top point in the round trip. In terms of θ, what is the coefficient of friction? draw a free body diagram for the sled going up the slope and another for the sled coming down the slope, show all work and explain in 1 - 2 sentences why it takes longer for the sled to slide down the slope than to go up the slope

Answer 1

Since we have push the sled therefore it has some initial velocity = u
Now consdering the FBD of case 1 when sled is moving upward
Ma = -(MgSin + f_K )
we know that f_K = u_K*R (where R is normal reaction )
f_K = u_K*MgCos, therefore
Ma =-( MgSin + u_K*MgCos )
a = -(gSin + u_K*gCos )
Now let us consider that the sled has travel distance along the incline = S
using kinematic equation
V² = u² +2aS
Where V is final velocity = 0
S = -(u²/2a)
We know that
V = u +at_up
u = -at_up
Putting the value in above equation
S = -(1/2)a*(t_up)² -----------(1)
Now considering the downward motion
In this case initial velocity will be (U) = 0
Using the FBD
MgSin - f_K = MA
MgSin- u_K*MgCos = MA
A = (gSin - u_KgCos)
Now using the kinematic equation
S = Ut_down +(1/2)A*t²_down = 0 +(1/2)A*t²_down
S = (1/2)A*t²_down
Now it is given that
t_down = 4*t_up
putting the value
S = (1/2)*A*(4*t_up)² = (1/2)A*16*t²_up ---------(2)
Equating 1 and 2
(1/2)A*16*t²_up = -(1/2)a*(t_up)²
16A = -a
Putting the value of both acceleration
16*g(Sin - u_KCos) = -(-(Sin + u_K*Cos ))*g
16Sin - 16u_KCos) = Sin + u_K*Cos
15Sin = 17u_KCos
u_K = (15/17)tan
Hence the coefficient of kinetic friction will be (15/17)tan
Now since we have provided the push in first case thats why the time taken to cover the distance was less.