super sam intends to jump off a 100 foot tower with an elastic cord tied to one ankle, ad the other end tied to the top of the tower. This cord is very light but very strong and stretches so that it can stop him without pulling his leg off. Such a cord exerts a force with the same mathematical form as the spring force. He wants it to be 75 feet long so that he will be in free fall for 75 feet before the cord begins to stretch. To minimize the force that the cord exerts on his leg, he wants it to stretch as far as possible. You have been assigned to purchase the cord for the stunt and must determine the elastic force constant which characterizes the cord that you should order. Before the calculation, you carefully measure sams height to be 6ft and his weight to be 170 lbs. For maxium dramatic effect, his jump will be off a diving board at the top of the tower. From tests you have made, you determine that his maximum speed coming off the diving board is 10 ft/sec. Neglect air resistance in your calculation.
To determine the elastic force constant needed for the cord, we will use the principles of conservation of energy.
Step 1: Calculate Sam's potential energy at the top of the tower.
The potential energy (PE) is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
Height of the tower (h) = 100 ft
Height of Sam (h_sam) = 6 ft
Weight of Sam (W) = 170 lbs
First, we need to convert Sam's weight from pounds to mass in pounds (lb) and then to kilograms (kg).
1 lb = 0.453592 kg
Weight (W) = m * g
170 lbs = m * 9.8 m/s^2
Converting pounds to kilograms:
170 * 0.453592 = 77.11044 kg
Step 2: Calculate Sam's weight at the top of the tower.
Since Sam is at the top of the tower, his weight is acting downward. Hence, his weight at the top of the tower is given by:
Weight at the top (W_top) = W - m_sam * g
where m_sam is Sam's mass and g is the acceleration due to gravity.
Weight at the top (W_top) = 77.11044 kg * 9.8 m/s^2
Step 3: Calculate Sam's potential energy at the top of the tower.
PE_top = m_sam * g * h
PE_top = 77.11044 kg * 9.8 m/s^2 * 100 ft
Step 4: Calculate Sam's final kinetic energy just before the cord starts to stretch.
Assuming no air resistance, Sam's total energy at the top of the tower is equal to his kinetic energy just before the cord starts to stretch.
KE = 1/2 * m_sam * v^2
Given:
Maximum speed (v) = 10 ft/s
Converting feet per second to meters per second:
1 ft/s = 0.3048 m/s
Maximum speed (v) = 10 ft/s * 0.3048 m/s
Step 5: Determine the initial length of the cord.
The initial length of the cord should be equal to the distance fallen before the cord starts to stretch. This is given as 75 ft.
Step 6: Calculate the stretching length of the cord.
To calculate the stretching length of the cord, we need to subtract the initial length of the cord from the total distance fallen.
Stretching length of the cord = Distance fallen - Initial length of the cord
= h_sam - initial length of the cord
= 75 ft - initial length of the cord
Step 7: Calculate the spring constant (k).
The spring constant (k) represents the stiffness of the cord.
The elastic potential energy stored in the stretched cord is given by the equation:
Elastic potential energy (PE_elastic) = (1/2) * k * x^2
Since the stretching length of the cord is the amount of stretch (x), we can substitute x = stretching length of the cord in the above equation.
PE_elastic = (1/2) * k * (stretching length of the cord)^2
Setting the potential energy at the top of the tower equal to the elastic potential energy just before the cord starts to stretch, we can equate the two equations.
PE_top = PE_elastic
Step 8: Substitute the values and solve for the spring constant (k).
PE_top = PE_elastic
m_sam * g * h = (1/2) * k * (stretching length of the cord)^2
Substituting the calculated values:
77.11044 kg * 9.8 m/s^2 * 100 ft = (1/2) * k * (75 ft)^2
Step 9: Convert the equation into SI units.
To maintain consistency in the units, we need to convert all the lengths to meters.
77.11044 kg * 9.8 m/s^2 * 30.48 m = (1/2) * k * (22.86 m)^2
Step 10: Solve for the spring constant (k).
(77.11044 kg * 9.8 m/s^2 * 30.48 m) / (1/2 * (22.86 m)^2) = k
Simplifying the equation will give you the value of the spring constant (k) that you should order for the cord.
To determine the elastic force constant of the cord that you should order, we can use the concept of Hooke's Law, which states that the force exerted by a spring or elastic cord is directly proportional to the displacement from its equilibrium position.
Let's break down the problem into steps:
Step 1: Determine the stretch of the cord
We are given that Super Sam wants the cord to be 75 feet long before it begins to stretch. Since his height is 6 feet, the total length of the cord needed will be 75 + 6 = 81 feet.
Step 2: Calculate the initial stretch of the cord
When Super Sam jumps off the tower, he will free fall for 75 feet before the cord begins to stretch. The initial stretch of the cord can be calculated by subtracting the initial length of the cord (81 feet) from the distance he falls before the cord starts to stretch (75 feet). Therefore, the initial stretch of the cord is 75 - 81 = -6 feet.
Step 3: Determine the maximum displacement of Super Sam during the jump
To minimize the force exerted on Super Sam's leg, we want the cord to stretch as far as possible. Based on the information provided, we know that Sam's maximum speed coming off the diving board is 10 ft/sec. To determine the maximum displacement, we can use the equation for uniform acceleration:
v^2 = u^2 + 2as
Where:
v = final velocity (0 ft/sec at maximum displacement)
u = initial velocity (10 ft/sec)
a = acceleration (due to gravity, approximately 32 ft/sec^2)
s = displacement (maximum displacement)
Rearranging the equation:
s = (v^2 - u^2) / (2a)
s = (0^2 - 10^2) / (2 * -32)
s = 100 / -64
s = -1.56 feet
The negative sign indicates that the displacement is in the opposite direction of the gravitational force.
Step 4: Calculate the force constant (k) of the cord
Now that we have the initial stretch (x = -6 feet) and the maximum displacement (s = -1.56 feet), we can use Hooke's Law to find the force constant (k) of the cord.
F = -kx
Where:
F = force exerted by the cord
x = initial stretch
Since the cord stretches further as it exerts a force, the force (F) must be negative in this case.
Using the given weight of Super Sam (170 lbs), we can convert it to mass:
Weight = mass * acceleration due to gravity
170 lbs = mass * 32 ft/sec^2
mass = 170 lbs / 32 ft/sec^2
mass ≈ 5.31 slugs (1 slug = 32.2 lbs * sec^2 / ft)
To convert the mass to SI units (kilograms):
mass ≈ 5.31 slugs * (14.59 kg/slug)
mass ≈ 77.39 kg
Next, let's convert the measurements to the appropriate units:
- Initial stretch (x) = -6 feet ≈ -1.83 meters
- Maximum displacement (s) = -1.56 feet ≈ -0.48 meters
Now, substituting the values into Hooke's Law:
F = -kx
mg = -kx
(77.39 kg) * (9.8 m/s^2) = -k * (-1.83 m)
Simplifying the equation:
k = (77.39 kg * 9.8 m/s^2) / (-1.83 m)
k ≈ 409.03 N/m
Therefore, you should order a cord with an elastic force constant (k) of approximately 409.03 N/m for Super Sam's stunt.
Note: This calculation neglects air resistance and assumes idealized conditions.