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4.5 Normal, tension, and other examples of force  (Page 8/11)

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Draw a free-body diagram to represent the forces acting on a kite on a string that is floating stationary in the air. Label the forces in your diagram.

The diagram has a black dot and three solid red arrows pointing away from the dot. Arrow Ft is long and pointing to the left and slightly down. Arrow Fw is also long and is a bit below a diagonal line halfway between pointing up and pointing to the right. A short arrow Fg is pointing down.

F g is the force on the kite due to gravity.

F w is the force exerted on the kite by the wind.

F t is the force of tension in the string holding the kite. It must balance the vector sum of the other two forces for the kite to float stationary in the air.

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A car is sliding down a hill with a slope of 20°. The mass of the car is 965 kg. When a cable is used to pull the car up the slope, a force of 4215 N is applied. What is the car’s acceleration, ignoring friction?

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Section summary

  • When objects rest on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called a normal force, N size 12{N} {} .
  • When objects rest on a non-accelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object:

    N = mg size 12{N= ital "mg"} {} .

  • When objects rest on an inclined plane that makes an angle θ size 12{θ} {} with the horizontal surface, the weight of the object can be resolved into components that act perpendicular ( w ⊥ ) and parallel ( w ∥ size 12{w rSub { size 8{ \lline \lline } } } {} ) to the surface of the plane. These components can be calculated using:

    w ∥ = w sin ( θ ) = mg sin ( θ ) size 12{w rSub { size 8{ \lline \lline } } =w"sin" \( θ \) = ital "mg""sin" \( θ \) } {}
    w ⊥ = w cos ( θ ) = mg cos ( θ ) size 12{w rSub { size 8{ ortho } } =w"cos" \( θ \) = ital "mg""cos" \( θ \) } {} .

  • The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension, T size 12{T} {} . When a rope supports the weight of an object that is at rest, the tension in the rope is equal to the weight of the object:

    T = mg size 12{T= ital "mg"} {} .

  • In any inertial frame of reference (one that is not accelerated or rotated), Newton’s laws have the simple forms given in this chapter and all forces are real forces having a physical origin.

Conceptual questions

If a leg is suspended by a traction setup as shown in [link] , what is the tension in the rope?

Diagram of a leg in traction.
A leg is suspended by a traction system in which wires are used to transmit forces. Frictionless pulleys change the direction of the force T without changing its magnitude.
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In a traction setup for a broken bone, with pulleys and rope available, how might we be able to increase the force along the tibia using the same weight? (See [link] .) (Note that the tibia is the shin bone shown in this image.)

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Problem exercises

Two teams of nine members each engage in a tug of war. Each of the first team’s members has an average mass of 68 kg and exerts an average force of 1350 N horizontally. Each of the second team’s members has an average mass of 73 kg and exerts an average force of 1365 N horizontally. (a) What is magnitude of the acceleration of the two teams? (b) What is the tension in the section of rope between the teams?

  1. 0. 11 m/s 2 size 12{0 "." "11 m/s" rSup { size 8{2} } } {}
  2. 1 . 2 × 10 4 N size 12{1 "." 2 times "10" rSup { size 8{4} } " N"} {}
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What force does a trampoline have to apply to a 45.0-kg gymnast to accelerate her straight up at 7 . 50 m/s 2 size 12{7 "." "50 m/s" rSup { size 8{2} } } {} ? Note that the answer is independent of the velocity of the gymnast—she can be moving either up or down, or be stationary.

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(a) Calculate the tension in a vertical strand of spider web if a spider of mass 8 . 00 × 10 − 5 kg size 12{8 "." "00" times "10" rSup { size 8{ - 5} } " kg"} {} hangs motionless on it. (b) Calculate the tension in a horizontal strand of spider web if the same spider sits motionless in the middle of it much like the tightrope walker in [link] . The strand sags at an angle of 12º size 12{"12"°} {} below the horizontal. Compare this with the tension in the vertical strand (find their ratio).

(a) 7 . 84 × 10 -4 N size 12{7 "." "84" times "10" rSup { size 8{4} } " N"} {}

(b) 1 . 89 × 10 –3 N size 12{1 "." "89" times "10" rSup { size 8{"–3"} } " N"} {} . This is 2.41 times the tension in the vertical strand.

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Suppose a 60.0-kg gymnast climbs a rope. (a) What is the tension in the rope if he climbs at a constant speed? (b) What is the tension in the rope if he accelerates upward at a rate of 1 . 50 m/s 2 size 12{1 "." "50 m/s" rSup { size 8{2} } } {} ?

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Show that, as stated in the text, a force F ⊥ size 12{F rSub { size 8{ ortho } } } {} exerted on a flexible medium at its center and perpendicular to its length (such as on the tightrope wire in [link] ) gives rise to a tension of magnitude T = F ⊥ 2 sin ( θ ) size 12{T= { {F rSub { size 8{ ortho } } } over {2"sin" \( θ \) } } } {} .

Newton’s second law applied in vertical direction gives

F y = F − 2 T sin θ = 0 size 12{F rSub { size 8{y} } =F - 2T" sin "θ=0} {}
F = 2 T sin θ size 12{F rSub { size 8{ ortho } } =2"T sin "θ} {}
T = F 2 sin θ size 12{T= { {F rSub { size 8{ ortho } } } over {"2 sin "θ} } } {} .

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Consider the baby being weighed in [link] . (a) What is the mass of the child and basket if a scale reading of 55 N is observed? (b) What is the tension T 1 size 12{T rSub { size 8{1} } } {} in the cord attaching the baby to the scale? (c) What is the tension T 2 size 12{T rSub { size 8{2} } } {} in the cord attaching the scale to the ceiling, if the scale has a mass of 0.500 kg? (d) Draw a sketch of the situation indicating the system of interest used to solve each part. The masses of the cords are negligible.

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A vertical spring scale measuring the weight of a baby is shown. The scale is hung from the ceiling by a cord. The weight W of the baby is shown by a vector arrow acting downward and tension T sub one acting in the cord is shown by an arrow upward. The tension in the cord T sub two attached to the ceiling is represented by an arrow upward from the spring scale and downward from the ceiling.
A baby is weighed using a spring scale.

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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