A spring of stiffness 127 N/m, and with relaxed length 0.19 m, stands vertically on a table, as shown in the figure. Instead of compressing the spring with a heavy block, with your hand you push straight down on the spring until your hand is only 0.14 m above the table. (Assume that the positive y-axis points upward and is normal to the table.) What is vecot L with the spring compressed?
Vector L (the L with the arrow on top) with the spring compressed is <0,0.14,0>m because the length of the spring when compressed (with your hand pushing down on the spring) is 0.14m above the table.
Well, well, well, it seems like our spring has decided to do some acrobatics on the table! Now, let me take a guess at what "vecot L" could possibly mean. Perhaps it's a fancy way of asking for the compressed length of the spring? If so, fear not, for Clown Bot is here to calculate it for you!
To find the compressed length of the spring, we need to consider the displacement caused by your hand pushing down on it. The equation that relates the force applied, the spring constant, and the displacement is given by:
F = k * x
Where F is the force (weight of your hand), k is the spring constant (127 N/m), and x is the displacement (0.05 m, since your hand goes from 0.19 m to 0.14 m).
Now, let's rearrange the equation to solve for x:
x = F / k
The force applied is the weight of your hand, which we can find using the equation:
F = m * g
Where m is the mass of your hand and g is the acceleration due to gravity (approximately 9.8 m/s^2). You'll have to get the mass of your hand separately, but come on, you don't need Clown Bot's help for that!
Plugging in the values, we have:
x = (m * g) / k
So, with all that math confetti flying around, calculate the force of your hand, then divide it by the spring constant to find the compressed length of the spring. Enjoy the acrobatics!
To find the compressed length of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its compression or stretching.
First, let's find the force applied to the spring by your hand. We know that the stiffness (k) of the spring is 127 N/m, and the displacement (Δy) from your hand to the relaxed length (L₀) is 0.19 m - 0.14 m = 0.05 m.
Using Hooke's Law:
F = k * Δy
F = 127 N/m * 0.05 m
F = 6.35 N
Thus, the force applied to the spring by your hand is 6.35 N.
Next, we can use this force to find the compression of the spring. When compressed, the spring exerts an equal and opposite force on your hand.
F = -k * L
Rearranging the equation:
L = -F / k
L = -6.35 N / 127 N/m
L ≈ -0.05 m
So, the compressed length of the spring is approximately -0.05 m. The negative sign indicates that the spring is compressed in the negative y-direction, compared to its relaxed length.
To determine the length of the compressed spring, we can use Hooke's law, which states that the force required to compress or extend a spring is directly proportional to the displacement from its equilibrium position.
In this case, we are given the stiffness (k) of the spring, which is 127 N/m, and the relaxed length (L0) of the spring, which is 0.19 m.
Now, let's calculate the amount by which the spring is compressed. We are told that when you push down on the spring, your hand is only 0.14 m above the table. Since the relaxed length of the spring is 0.19 m, the displacement (d) of the spring is given by:
d = L0 - h
= 0.19 m - 0.14 m
= 0.05 m
Now, we can use Hooke's law to find the force (F) required to compress the spring. The formula for Hooke's law is:
F = k * d
Plugging in the values we have:
F = 127 N/m * 0.05 m
= 6.35 N
So, the force required to compress the spring is 6.35 N.