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A car of mass 1500 kg is lifted up a dis...

A car of mass 1500 kg is lifted up a distance of 30 m by crane A in 0.5 minutes. The second crane B does the same jod in 1 minute . The ratio of their powers is

A

`1:2`

B

`2:1`

C

`1:4`

D

`4:1`

Text Solution

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The correct Answer is:
To solve the problem of finding the ratio of powers of two cranes lifting a car, we can follow these steps: ### Step-by-Step Solution: 1. **Identify the Given Data:** - Mass of the car (m) = 1500 kg - Distance lifted (h) = 30 m - Time taken by Crane A (t_A) = 0.5 minutes = 0.5 × 60 seconds = 30 seconds - Time taken by Crane B (t_B) = 1 minute = 60 seconds 2. **Calculate the Work Done:** - The work done (W) in lifting the car is equal to the change in potential energy (PE), which can be calculated using the formula: \[ W = mgh \] - Where \( g \) (acceleration due to gravity) is approximately \( 9.81 \, \text{m/s}^2 \). - Therefore, substituting the values: \[ W = 1500 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 30 \, \text{m} \] \[ W = 1500 \times 9.81 \times 30 = 441450 \, \text{Joules} \] 3. **Calculate the Power for Each Crane:** - Power (P) is defined as work done per unit time: \[ P = \frac{W}{t} \] - For Crane A: \[ P_A = \frac{W}{t_A} = \frac{441450 \, \text{J}}{30 \, \text{s}} = 14715 \, \text{W} \] - For Crane B: \[ P_B = \frac{W}{t_B} = \frac{441450 \, \text{J}}{60 \, \text{s}} = 7374.5 \, \text{W} \] 4. **Calculate the Ratio of Powers:** - The ratio of the powers of Crane A to Crane B is: \[ \text{Ratio} = \frac{P_A}{P_B} = \frac{14715}{7374.5} \] - Simplifying this gives: \[ \text{Ratio} = 2 \] - Therefore, the ratio of their powers is: \[ P_A : P_B = 2 : 1 \] ### Final Answer: The ratio of the powers of Crane A to Crane B is \( 2 : 1 \).
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