What should be the spring constant k (in N/m) of a spring designed to bring a 1105-kg car to rest from a speed of 69-km/h so that the occupants undergo a maximum acceleration of 3.8g's?
F = k x
potential energy in spring = .5 k x^2
kinetic energy in car = .5 m v^2
so in the end
k Xmax^2 = m Vi^2 = 1105 (69000/3600)^2
Fmax = m Amax
k Xmax = 1105 (3.8*9.81)
so
Xmax = 1105(3.8*9.81)/k = 41192/k
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k(1.7*10^9)/k^2 = 406,000 =4.06*10^5
4.06 k = 1.7*10^4
k = 4187 N/m
check my arithmetic
To determine the spring constant (k) of a spring designed to bring a car to rest while subjecting the occupants to a maximum acceleration of 3.8g's, we can follow these steps:
Step 1: Convert the speed of the car from km/h to m/s.
Given: Speed of the car = 69 km/h
1 km/h = 0.2778 m/s (approx.)
Converting from km/h to m/s:
Speed of the car = 69 km/h × 0.2778 m/s = 19.17 m/s (approx.)
Step 2: Calculate the deceleration (a) required to bring the car to rest.
Given: Initial velocity (u) = 19.17 m/s
Final velocity (v) = 0 m/s (since the car comes to rest)
Time taken (t) = ?
Acceleration (a) = ?
Using the equation: v = u + at
0 = 19.17 m/s + a(t)
Step 3: Convert the maximum acceleration from g's to m/s².
Given: Maximum acceleration = 3.8g's
1g = 9.8 m/s² (approx.)
Converting from g's to m/s²:
Maximum acceleration = 3.8 × 9.8 m/s² ≈ 37.24 m/s²
Step 4: Equate the maximum acceleration to the acceleration calculated in Step 2.
37.24 m/s² = a
Step 5: Solve the equation from Step 2 to find the time taken (t).
0 = 19.17 m/s + 37.24 m/s² × t
Solving for 't', we get:
t = - 19.17 m/s ÷ (37.24 m/s²)
t ≈ -0.5159 s
Note: The negative sign indicates that the car is decelerating.
Step 6: Calculate the spring constant (k) using Hooke's law.
Given: Mass of the car (m) = 1105 kg
Using the equation: F = kx
Where F = Force, k = Spring constant, and x = Displacement
The force required to bring the car to rest is given by Newton's second law:
F = ma
Since the acceleration is negative and equal to (-37.24 m/s²):
F = (-37.24 m/s²) × (1105 kg)
F ≈ -41154.2 N
The force exerted by the spring is proportional to the displacement (x) of the spring from its equilibrium position. To calculate the displacement of the spring, we can use the equation:
F = kx
Solving for 'k':
k = F / x
But we know that F = -41154.2 N, so the equation becomes:
k = -41154.2 N / x
Note: The minus sign indicates that the force exerted by the spring is in the opposite direction to the displacement.
To find 'x,' we can use the equation for displacement:
x = ut + (1/2)at²
Substituting the known values:
x = (19.17 m/s) × (-0.5159 s) + (1/2) × (-37.24 m/s²) × (-0.5159 s)²
x ≈ -4.95 m
Substituting the values of F = -41154.2 N and x = -4.95 m into the equation for 'k':
k = (-41154.2 N) / (-4.95 m)
k ≈ 8323.2 N/m
Therefore, the spring constant k for the spring designed to bring the car to rest while subjecting the occupants to a maximum acceleration of 3.8g's is approximately 8323.2 N/m.
To find the spring constant (k) of the spring, we need to use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Hooke's Law can be expressed as:
F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement of the spring from its equilibrium position.
In this case, the spring is being used to bring the car to rest from a certain speed, so we want to find k.
Here's how we can approach this problem step by step:
Step 1: Convert the speed from km/h to m/s.
Given: Speed = 69 km/h
To convert to m/s, we divide by 3.6 (since 1 km/h = 0.2778 m/s).
Speed = 69 km/h × (1 m/s / 3.6 km/h)
Speed = 19.17 m/s
Step 2: Determine the maximum acceleration in m/s².
Given: Maximum acceleration = 3.8g's
To convert from g's to m/s², multiply by the acceleration due to gravity (g = 9.8 m/s²).
Maximum acceleration = 3.8 × 9.8 m/s²
Maximum acceleration = 37.24 m/s²
Step 3: Calculate the force required to bring the car to rest.
Using Newton's second law, force (F) is defined as the product of mass (m) and acceleration (a).
Force = mass × acceleration
Force = 1105 kg × (-37.24 m/s²) [Note: The force is negative because it opposes the motion.]
Force = -41030.2 N
Step 4: Determine the displacement (x) of the spring.
The displacement of the spring can be calculated using the formula:
Force = -kx
Since we know the force and the maximum acceleration (which is proportional to the force), we can write:
-41030.2 N = -k × x
Step 5: Solve for the spring constant (k).
Rearranging the equation, we have:
k = Force / x
Substituting the values we have:
k = -41030.2 N / x
Step 6: Determine the displacement of the spring.
The displacement of the spring (x) can be calculated by using the formula:
Speed² = (2 × acceleration × displacement)
Rearranging the formula, we get:
Displacement (x) = (Speed²) / (2 × acceleration)
Substituting the values:
Displacement (x) = (19.17 m/s)² / (2 × 37.24 m/s²)
Displacement (x) = 19.17² m²/s² / (2 × 37.24)
Displacement (x) ≈ 1.571 m²/s²
Step 7: Calculate the spring constant (k).
Now we can substitute the value of displacement (x) into the formula for k:
k = -41030.2 N / (1.571 m²/s²)
k ≈ -26131.57 N/m
The spring constant (k) for the spring designed to bring the car to rest from a speed of 69 km/h, so that the occupants undergo a maximum acceleration of 3.8g's, is approximately -26131.57 N/m. The negative sign indicates that the spring is stretched (displaced from its equilibrium position) when it exerts the required force to bring the car to rest.