A 1200-kg car is being raised with a constant acceleration of 2.53 m/s² by a crane,using a 20-m long steel cable that is 1.5 cm in diameter.Young's modulus for steel is 2.0 × 10¹¹ N/m².What is the change in length of the cable caused by lifting the car? A)8.4 mm B)8.4 cm C)4.9 mm D)4.9 cm E)4.9 m

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Quizplus · Accepted Answer

The change in length ($\Delta L$) of the cable can be calculated using the formula $\Delta L = \frac{FL}{A E}$, where $F$ is the force applied, $L$ is the original length of the cable, $A$ is the cross-sectional area of the cable, and $E$ is Young's modulus for the material of the cable. The force $F$ includes both the gravitational force on the car and the force due to the acceleration of the car upwards. The gravitational force is $mg$, where $m$ is the mass of the car and $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$). The additional force due to the car's acceleration is $ma$, where $a$ is the acceleration of the car. Thus, $F = mg + ma$. The cross-sectional area $A$ of the cable can be calculated using the formula for the area of a circle, $A = \pi r^2$, where $r$ is the radius of the cable. Given that the diameter of the cable is $1.5 \, \text{cm}$, the radius is $0.75 \, \text{cm}$ or $0.0075 \, \text{m}$. Substituting the given values and calculating gives a change in length of $8.4 \, \text{mm}$.

A 1200-Kg Car Is Being Raised with a Constant Acceleration