A dart is loaded into a spring-loaded toy dart gun by pushing the spring in by a distance d. For the next loading?the spring is compressed a distance d/3. How much work is required to load the second dart compared to that required to load the first?
A)nine times as much
B)three times as much
C)the same
D)one-third as much
E)one-Ninth as much
work=1/2 k x^2 so if x is 1/3, energy must be 1/9.
To determine the work required to load the second dart compared to the first, we can use the concept of work done by a spring.
The work done by a spring is given by the formula:
W = (1/2)kx^2
Where W is the work done, k is the spring constant, and x is the distance the spring is compressed or extended.
We are given that the spring is compressed a distance d to load the first dart. Therefore, the work required to load the first dart would be:
W1 = (1/2)kd^2
For the second loading, the spring is compressed a distance d/3. Therefore, the work required to load the second dart would be:
W2 = (1/2)k(d/3)^2
To compare the work required for the second dart to the first dart, we can compare the ratios of their respective work done:
W2/W1 = [(1/2)k(d/3)^2] / [(1/2)kd^2]
= [(1/2)(1/9)kd^2] / [(1/2)kd^2]
= (1/9)
Therefore, the work required to load the second dart is one-ninth (1/9) as much as the work required to load the first dart.
The correct answer is E) one-Ninth as much.
To determine the amount of work required to load the second dart compared to the first, we need to consider the relationship between the work done and the displacement of the spring.
The work done on an object can be calculated using the formula:
Work (W) = Force (F) × Displacement (d) × cos(θ)
In this case, the force required to compress the spring is directly proportional to the displacement. If we let the force required to compress the spring by distance d be F1, then the force required to compress it by distance d/3 would be F2 = (1/3) × F1.
Now, let's compare the work required for loading the first dart (W1) to the work required for loading the second dart (W2):
W1 = F1 × d (work for loading first dart)
W2 = F2 × (d/3) (work for loading second dart)
Substituting the value of F2 in terms of F1:
W2 = [(1/3) × F1] × (d/3) = (d/9) × F1
We can see that the work required to load the second dart (W2) is proportional to the displacement (d) divided by 9 times the force required to compress the spring (F1).
Therefore, the work required to load the second dart compared to the first is one-ninth as much. Hence, the answer is (E) one-ninth as much.