Question

Please show all of your work. Also, please draw a sketch with a coordinate system and important points labeled.

Moreover, I need appropriate graphs

A box is released from rest at the top of a ramp. The surface of the ramp makes an angle of 31.5degrees with the horizontal, and is rough; the coefficients of friction between the ramp and the box are js = 0.500 and Mik = 0.350. What is the magnitude of the acceleration of the box?

Answer 1

Free Body diagram is given below:

ч W. Coso W. sing w

By force balance in the direction of inclined plane,

W*sin $\theta$ + Fk = m*a eq(1)

here, W = weight = m*g

m = mass of box

g = 9.81 /s^2

$\theta$ = angle of inclined plane = 31.5 deg

a = acceleration of box = ??

Fr = friction force applied on box = -$\mu$*N

N = normal reaction force = ??

given, $\mu$ = Co-efficient of kinetic friction = 0.350

from force balance in the perpendicular direction of inclined plane,

N - W*cos$\theta$ = 0 (since there is no acceleration in the perpendicular direction of inclined plane.)

N = m*g*cos$\theta$

So, from eq(1),

m*g*sin$\theta$ - $\mu$ *m*g*cos$\theta$ = m*a

a = g*sin$\theta$ - $\mu$ *g*cos$\theta$

a= 9.81*sin(31.5 deg) - 0.350*9.81*cos(31.5 deg) = 2.198 m/s^2

a = 2.20 m/s^2

Let me know if you have any query. (And there is no requirement of graph for anything)