Question

An object of mass 3kg rest on a plane. The coefficient of static friction and that of kinetic friction are given by Us=0.3 and uk= 0.2. The plane inclined at an angle theta to the horizontal. (a) Find the maximum value of the theta for which the object remains at rest. (b) find the acceleration of the object if it started sliding from rest down the plane at the angle theta maximum to the horizontal. (c) how long does it take the object to move from rest, a distance of 1m under the condition of B

Answer 1

(a) at equilibrium,

static friction = component of weight along plane .

or, $\mu_sN=mgsin\theta$

N = component of weight perpendicular to plane.

so, $\mu_s mgcos\theta=mgsin\theta$

or, $\mu_s= 0.3=tan\theta$

hence, $\theta=tan^{-1}(0.3)$

(b) when object starts to move, kinetic friction applied on it.

from Newton's 2nd law,

Fnet = ma

or, $mgsin\theta-\mu_kN=ma$

or, $mgsin\theta-\mu_kmgcos\theta=ma$

or, $gsin\theta-\mu_kcos\theta=a$

or, $a=gsin[tan^{-1}(0.3)]-0.2gcos[tan^{-1}(0.3)]$

or, a = g × 3/√(109) - 0.2g × 10/√(109)

= g/√(109) m/s² ≈ 0.96 m/s²

(c) s = ut + 1/2 at²

or, 1m = 0 + 1/2 × 0.96 t²

or, 1/0.48 = t²

or, t = √(2.08) = 1.44 sec