Equilibrium of the bodies on a inclined plane – application of the Newton’s first law problems and solutions

1. A 2-kg block lies on a rough inclined plane at an angle 37^o to the horizontal. Determine the magnitude of the external force exerted on the block, so the block is not slides down the plane. (sin 37^o = 0.6, cos 37^o = 0.8, g = 10 m.s^-2, µ_k = 0.2)

Known :

Mass (m) = 2 kg

Acceleration due to gravity (g) = 10 m/s²

Block’s weight (w) = m g = (2)(10) = 20 Newton

Sin 37^o = 0.6

Cos 37^o = 0.8

Coefficient of the kinetic friction (µ_k) = 0.2

The y-component of the weight (w_y) = w cos 37^o = (20)(0.8) = 16 Newton

The x-component of the weight (w_x) = w sin θ = (20)(sin 37) = (20)(0.6) = 12 Newton

the normal force (N) = w_y = 16 Newton

Wanted : The external force (F)

Solution :

w_x = 12 Newton

The force of the kinetic friction (f_k) = µ_k N = (0.1)(16) = 1.6 Newton

The magnitude of the external force F exerted on the block :

F + f_k – w_x = 0

F = w_x – f_k

F = 12 – 1.6

F = 10.4 Newton

The external force F greater than 10.4 Newton.

2. Mass of a block = 2 kg, coefficient of static friction µ_s = 0.4 and θ = 45^o. Determine the magnitude of the force F so the block start to slides up.

Known :

The coefficient of the static friction (µ_s) = 0.4

Angle (θ) = 45^o

Acceleration due to gravity (g) = 10 m/s²

Block’s mass (m) = 2 kilogram

Block’s weight (w) = m g = (2 kg)(10 m/s²) = 20 kg m/s² = 20 Newton

The x-component of the weight (w_x) = w sin θ = (20)(sin 45) = (20)(0.5√2) = 10√2 Newton

The y-component of the weight (w_y) = w cos θ = (20)(cos 45) = (20)(0.5√2) = 10√2 Newton

Wanted : The magnitude of the force F

Solution :

Block starts to slide up, if F ≥ w_x + f_s.

The x-component of the weight :

w_x = 10√2 Newton

the y-component of the weight :

w_y = 10√2 Newton

The normal force :

N = w_y = 10√2 Newton

The force of the static friction :

f_s = µ_s N = (0,4)(10√2) = 4√2

The magnitude of the force F so that the block starts to slide up :

F ≥ w_x + f_s

F ≥ 10√2 + 4√2

F ≥ 14√2 Newton

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1. Particles in one-dimensional equilibrium
2. Particles in two-dimensional equilibrium
3. Equilibrium of bodies connected by cord and pulley
4. Equilibrium of bodies on the inclined plane