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Chapter 5: Problem 3

A lamp hangs vertically from a cord in a descending elevator. The elevator has a deceleration of \(2.4 \mathrm{~m} / \mathrm{s}^{2}\) before coming to a stop. ( \(a\) ) If the tension in the cord is \(89 \mathrm{~N}\), what is the mass of the lamp? (b) What is the tension in the cord when the elevator ascends with an upward acceleration of \(2.4 \mathrm{~m} / \mathrm{s}^{2}\) ?

Short Answer

Expert verified
The lamp has a mass of \(7.2 kg\) and when the lift moves upwards, the tension in the cord becomes \(87.84 N\).

Step by step solution

01

Identifying the given quantities and unknown

You have the tension \(T = 89 N\) in the cord and the acceleration due to the elevator's motion \(a = 2.4 m/s^2\). The unknown we're trying to find is the mass \(m\) of the lamp.
02

Apply Newton's second law to find mass

Newton's second law states that the net force on an object is equal to the mass of the object times the acceleration of the object. In the case of the descending elevator, the net force is the difference between the tension and the weight of the lamp, which gives \(T - mg = -ma\) where \(g\) is the acceleration due to gravity. Solving for \(m\) gives \(m = T / (g + a)\).
03

Calculations for (a)

Substitute the given values \(T = 89 N, g = 9.8 m/s^2, a = 2.4 m/s^2\) into the equation from step 2 to find \(m = 89 N/ (9.8 m/s^2 + 2.4 m/s^2) = 7.2 kg\).
04

Finding the tension in upward moving elevator

In the upward moving elevator, the tension acts in the opposite direction to the weight of the object. This gives \(T - mg = ma\), where \(T\) is the tension, \(m\) is the mass of the lamp, \(g\) is acceleration due to gravity, and \(a\) is the acceleration of the elevator. Do the math to calculate the new tension.
05

Calculations for (b)

Substitute the calculated mass \(m = 7.2 kg\), acceleration due to gravity \(g = 9.8 m/s^2\), acceleration of the elevator \(a = 2.4 m/s^2\) in the equation to get \(T = m * (g + a) = 7.2 kg * (9.8 m/s^2 + 2.4 m/s^2) = 87.84 N\).

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