A 2.14 kg block is held in equilibrium on an incline of angle q = 67.7o by a horizontal force, F.
If the coefficient of static friction between block and incline is ms = 0.314, determine the minimum value of F.
Determine the normal force of the incline on the block.
x: m•g•sinα-F(fr) –F•cosα = 0
y: N-m•gvcosα –Fvsinα = 0.
N = m•gvcosα +F•sinα,
F(fr) = μ•N=μ• (m•gvcosα +F•sinα).
m•g•sinα - μ• (m•g•cosα +Fvsinα) – F•cosα = 0
m•g•sinα - μ•m•g•cosα + μ•F•sinα – F•cosα = 0
F• (cosα – μ•sinα) =m•g• (sinα – μ•cosα)
F= m•g• (sinα – μ•cosα)/ (cosα – μ•sinα)=
= 2.14•9.8•(0.93- 0.314•0.38)/0.38-0.314•0.93) =
=20.97(0.81)/0.088=193.06 N.
N = m•gvcosα +F•sinα =
=2.14•9.8•0.38 + 193.16•0.93 =
=187.52 N
To determine the minimum value of the horizontal force F that keeps the block in equilibrium, we need to consider the forces acting on the block.
1. Weight:
The weight of the block can be calculated using the formula W = m*g, where m is the mass of the block and g is the acceleration due to gravity (9.8 m/s^2).
W = 2.14 kg * 9.8 m/s^2 = 20.972 N
2. Normal force:
The normal force is the force exerted by a surface perpendicular to the object. In this case, it is the force exerted by the incline on the block. Since the block is in equilibrium, the normal force is equal to the component of weight perpendicular to the incline.
Normal force = W * cos(q) = 20.972 N * cos(67.7°) = 8.874 N
3. Friction force:
The maximum static friction force is given by the equation F_friction = µ_s * N, where µ_s is the coefficient of static friction and N is the normal force.
Friction force = 0.314 * 8.874 N = 2.783 N
4. Horizontal force F:
In order to keep the block in equilibrium, the horizontal force F must balance the friction force. Since the block is at the minimum value of F, the force required to overcome friction is exactly equal to the maximum static friction force.
F = Friction force = 2.783 N
Therefore, the minimum value of F required to keep the block in equilibrium is 2.783 N.
To summarize:
- Minimum value of F = 2.783 N
- Normal force = 8.874 N