This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Calculate the weight of the body and its components parallel and perpendicular to the inclined plane. Given: mass m = 12 kg, angle of inclination = 30^, coefficient of friction = (2)/(3), and g = 10 ms^-2. The weight of the body is W = mg. W = 12 kg × 10 ms^-2 = 120 N The component of weight parallel to the plane (acting down the plane) is W_. W_ = W = 120 30^ = 120 × (1)/(2) = 60 N The component of weight perpendicular to the plane is W_. W_ = W = 120 30^ = 120 × sqrt(3)2 = 60sqrt(3) N The normal reaction force R is equal to W_. R = 60sqrt(3) N The maximum static frictional force F_f is R. F_f = (2)/(3) × 60sqrt(3) = 40sqrt(3) N Note that W_ = 60 N and F_f = 40sqrt(3) N ≈ 40 × 1.732 = 69.28 N. Since W_ < F_f, the body will not slide down the plane on its own. a) Find the value of P if the body is at the point of moving down the plane. Step 2: Determine the direction of P and set up the force balance equation for impending downward motion. Since the body would not slide down on its own (W_ < F_f), the external force P must be acting down the plane to initiate motion. At the point of impending motion down the plane, the forces acting down the plane are balanced by the forces acting up the plane. Friction acts up the plane, opposing the downward motion. P + W_ = F_f Substitute the calculated values: P + 60 N = 40sqrt(3) N Solve for P: P = (40sqrt(3) - 60) N The value of P is (40sqrt(3) - 60) N. b) Find the value of P if the body is at the point of moving up the plane. Step 3: Determine the direction of P and set up the force balance equation for impending upward motion. For the body to move up the plane, the external force P must be acting up the plane. At the point of impending motion up the plane, the forces acting up the plane are balanced by the forces acting down the plane. Friction acts down the plane, opposing the upward motion. P = W_ + F_f Substitute the calculated values: P = 60 N + 40sqrt(3) N P = (60 + 40sqrt(3)) N The value of P is (60 + 40sqrt(3)) N. Last free one today — make it count tomorrow, or type /upgrade for unlimited.