A steel cable of cross sectional area 2.5 cm^2 supports a 1000kg elevator. The elastic limit of the cable is 3 x 10^8 Pa. If the stress in the cable is not to exceed 20 percent of the elastic limit, then find the maximum upward acceleration of the elevator
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To find the maximum upward acceleration of the elevator, we need to calculate the maximum allowable tension in the steel cable.
1. Calculate the maximum allowable stress in the cable:
The stress (σ) in the cable can be calculated using the formula:
σ = F / A
where σ is the stress, F is the force, and A is the cross-sectional area of the cable.
Given:
Force (F) = weight of the elevator = mass (m) x acceleration due to gravity (g)
mass (m) = 1000 kg
acceleration due to gravity (g) = 9.8 m/s^2
cross-sectional area (A) = 2.5 cm^2 = (2.5 / 10000) m^2 (since 1 cm^2 = (1 / 10000) m^2)
Substituting the values:
σ = (1000 kg x 9.8 m/s^2) / (2.5 / 10000) m^2
2. Calculate the maximum allowable tension in the cable:
The tension in the cable can be calculated by multiplying the stress by the cross-sectional area of the cable.
T = σ x A
Substituting the calculated value of σ and A:
T = (1000 kg x 9.8 m/s^2) / (2.5 / 10000) m^2 x (2.5 / 10000) m^2
3. Calculate the maximum upward acceleration:
The maximum upward acceleration can be calculated by dividing the maximum allowable tension by the weight of the elevator.
a = T / m
Substituting the calculated value of T and m:
a = [(1000 kg x 9.8 m/s^2) / (2.5 / 10000) m^2] / 1000 kg
4. Calculate the maximum upward acceleration:
Simplify the expression to find the value of maximum upward acceleration (a).
After performing the calculations, the maximum upward acceleration of the elevator can be determined by evaluating the given expression.