# Equilibrium of bodies connected by cord and pulley – application of Newton’s first law problems and solutions

1. A box of mass 5 kg is on an inclined plane at an angle 30^{o}. The box supported by a cord. Determine the tension force (T) and the normal force (N)!

Solution

∑F_{x}

T – w sin 30^{o} = 0

T = w sin 30^{o }

T = (5 kg)(9.8 m/s^{2}) sin 30^{o }

T = (49)(0.5)

T = 24.5 Newton

∑F_{y} = 0

N – w cos 30^{o} = 0

N = w cos 30^{o }

N = (49)(0.87)

N = 43 Newton

2. Two objects of mass m_{1} = m_{2} = 2 kg, connected by a massless string over a frictionless pulley. Find the tension force T_{1} and T_{2}.

Solution

(a) Free-body diagram for object 1 (b) Free-body diagram for object 2

Apply Newton’s first law to object 1 :

∑F_{y} = 0

T_{1} – w_{1} = 0

T_{1} = w_{1} = m_{1 }g = (2 kg)(9.8 m/s^{2}) = 19.6 N

Apply Newton’s first law to object 2 :

∑F_{y} = 0

T_{2} – w_{2} = 0

T_{2} = w_{2 }= m_{2} g = (2 kg)(9.8 m/s^{2}) = 19.6 N

T_{1} = T_{2 }= 19.6 N.

3. An object of weight w_{A} = 30 N and an object of weight w_{B} = 40 N, are attached by a lightweight cord that passes over a frictionless pulley of the negligible mass. Determine the coefficient of the maximum static friction between w_{B} and inclined surface, if the system is at rest.

Solution

(a) Free-body diagram for object w_{A} (b) Free-body diagram for object w_{B}

Apply Newton’s first law to object w_{A} in vertical (y) direction :

∑F_{y} = 0 (no acceleration in vertical direction)

T – w_{A }= 0

T = w_{A} = 30 Newton

Apply Newton’s first law to object w_{B }in vertical (y) direction :

∑F_{y} = 0

N – w_{B} cos 45^{o }= 0

N = w_{B }cos 45^{o }= (40)(0.7) = 28 Newton

Apply Newton’s first law to object w_{B }in horizontal (x) direction :

∑F_{x }= 0

F_{k} + w_{B} sin 45^{o} – T = 0

μ_{s} N + w_{B} sin 45^{o} – T = 0

μ_{s} (28) + (40)(0.7) – 30 = 0

μ_{s} (28) + 28 – 30 = 0

μ_{s }(28) = 30 – 28

μ_{s }(28) = 2

μ_{s }= 2 / 28

μ_{s }= 0.07

The coefficient of the maximum static friction between w_{B} and inclined surface = 0.07.

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- Particles in one-dimensional equilibrium
- Particles in two-dimensional equilibrium
- Equilibrium of bodies connected by cord and pulley
- Equilibrium of bodies on inclined plane