Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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1310001     The net force on a dart in a spring-loaded dart gun.     1310001
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Question

 

A spring-loaded dart gun is used to shoot a 5-gram dart straight upward. Its spring has a stiffness 50 N/m, and its relaxed length is 5.0 cm. The spring is compressed 4.0 cm from its relaxed length.

  1. What is the net force on the dart at the instant the spring is released?
  2. What is the net force on the dart at the instant the dart leaves the spring? At this instant, the spring has returned to its relaxed length.

 



Solution

 

Begin by drawing a picture, showing the spring and dart. Define the coordinate system and indicate the compression of the spring.


Figure: A compressed spring and dart in a loaded dart gun.

(a) At the instant the spring is released, it is compressed 0.04 m from its relaxed length. A free-body diagram for the dart at this instant is shown below.


Figure: A free-body diagram for the dart when the spring is fully compressed.

The net force on the dart is the sum of the forces on the dart. The net force on the dart can be found by adding the vectors tail to head from the free-body diagram, and by summing the forces algebraically. The algebraic answer should be consistent with the picture, as shown below.


Figure: A free-body diagram for the dart.

The spring force is given by Hooke's law:

In this case, the spring has a stiffness and is compressed 0.04 m from its relaxed length and is directed upward. Substitute the spring force and gravitational force on the dart and solve for the net force.

The net force is upward which is consistent with our free-body diagram and our picture showing the sum of the forces on the dart.

(b) At the instant the dart leaves the spring, the spring is at its relaxed length. In this case, the only force acting on the dart is the gravitational force on the dart.

A picture of the dart and a free-body diagram at the instant the dart leaves the spring are shown below. The free-body diagram for the dart is consistent with our calculation.


Figure: A free-body diagram for the dart when it leaves the spring.