The innerspring mattress on your grand- mother’s bed is held up by 23 vertical springs, each having a spring constant of 6000 N/m. A 43 kg person jumps from a 1.93 m platform onto the innersprings.
The acceleration of gravity is 9.8 m/s2 .
Assume: The springs were initially un- stretched and that they stretch equally (typi- cal old fashioned bed).
Determine the stretch of each of the springs. Answer in units of m
To determine the stretch of each of the springs, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement.
Step 1: Calculate the initial gravitational potential energy of the person before jumping.
The initial gravitational potential energy can be calculated using the formula: PE = m * g * h,
where m is the mass of the person (43 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the platform (1.93 m).
PE = 43 kg * 9.8 m/s^2 * 1.93 m
Step 2: Calculate the total force exerted by the person on the springs upon landing.
The total force exerted by the person can be calculated using the formula: F = m * a,
where m is the mass of the person (43 kg), and a is the acceleration due to gravity (9.8 m/s^2).
F = 43 kg * 9.8 m/s^2
Step 3: Calculate the stretch of each spring.
Each spring is being stretched by an equal amount, so we divide the total force by the number of springs to find the force exerted on each spring.
Force exerted on each spring = Total force / Number of springs
Force exerted on each spring = F / 23
Finally, we can use Hooke's Law to determine the stretch of each spring.
The formula for Hooke's Law is: F = k * x,
where F is the force exerted by the spring, k is the spring constant (6000 N/m), and x is the displacement or stretch of the spring.
Rearranging the formula, we get: x = F / k
Step 4: Substitute the values and calculate the stretch of each spring.
Stretch of each spring = Force exerted on each spring / Spring constant
Stretch of each spring = (F / 23) / 6000 N/m
Now, you can plug in the values and do the calculations to find the stretch of each spring.
To determine the stretch of each spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed.
First, let's calculate the total force exerted by the person on the springs. This can be done by calculating the weight of the person using the formula:
Weight (W) = mass (m) × acceleration due to gravity (g)
W = 43 kg × 9.8 m/s^2
W = 421.4 N
Since there are 23 springs in total, we can distribute the force evenly among them. Therefore, the force exerted on each spring is:
Force per spring = Total force / Number of springs
Force per spring = 421.4 N / 23
Force per spring ≈ 18.32 N
Now, using Hooke's Law, we can find the stretch of each spring using the formula:
Force = spring constant × stretch
Rearranging the formula, we get:
Stretch = Force / spring constant
Stretch = 18.32 N / 6000 N/m
Stretch ≈ 0.003053 m
Therefore, the stretch of each of the springs is approximately 0.003053 meters (or 3.053 mm).
PE-> KE -> PE(springs)
mgh =23 kx²/2
x=sqrt{2mgh/23k} =
=sqrt{2•43•9.8•1.93/23•6000} =0.012 m