Question

1. one of the tallest radio towers is in fargo, north dakota. the tower is 629.9 m tall. if a bird lands on top of the tower, so that the gravitational potential energy associated with the bird is 2033.76 j, what is its mass, in kilograms?

Explanation:

Step1: Recall the formula for gravitational potential energy

The formula for gravitational potential energy is \( PE = mgh \), where \( PE \) is the potential energy, \( m \) is the mass, \( g \) is the acceleration due to gravity (we'll use \( g = 9.8 \, \text{m/s}^2 \) as the standard value), and \( h \) is the height. We need to solve for \( m \), so we can rearrange the formula to \( m=\frac{PE}{gh} \).

Step2: Substitute the given values into the formula

We are given \( PE = 2033.76 \, \text{J} \), \( h = 629.9 \, \text{m} \), and \( g = 9.8 \, \text{m/s}^2 \). Plugging these values into the formula for \( m \):

\( m=\frac{2033.76}{9.8\times629.9} \)

First, calculate the denominator: \( 9.8\times629.9 = 9.8\times629.9 = 6173.02 \)

Then, divide the numerator by the denominator: \( m=\frac{2033.76}{6173.02}\approx0.3295 \) (we can also do the calculation more precisely)

Wait, let's check the calculation again. Wait, maybe I made a mistake in the multiplication. Let's recalculate \( 9.8\times629.9 \):

\( 629.9\times9.8 = 629.9\times(10 - 0.2)=629.9\times10-629.9\times0.2 = 6299 - 125.98 = 6173.02 \). Then \( 2033.76\div6173.02\approx0.3295 \). Wait, but let's check if the height is 629.9 m, potential energy 2033.76 J. Wait, maybe I messed up the formula? Wait, no, gravitational potential energy is \( PE = mgh \), so solving for m: \( m = PE/(gh) \). Let's compute it again:

\( gh = 9.8\times629.9 = 6173.02 \)

\( m = 2033.76 / 6173.02 \approx 0.3295 \) kg? Wait, that seems small, but let's check the numbers. Wait, 2033.76 divided by 6173.02. Let's do this division:

2033.76 ÷ 6173.02 ≈ 0.3295. Wait, maybe the height is 62.99 m? Wait, the problem says 629.9 m. Wait, maybe a typo? But according to the problem, we have to use 629.9 m. Wait, or maybe I used the wrong g? Let's try g = 9.81.

Let's recalculate with g = 9.81:

\( gh = 9.81\times629.9 = 9.81\times629.9 \)

Calculate 629.9×9 = 5669.1, 629.9×0.81 = 510.219, so total is 5669.1 + 510.219 = 6179.319

Then \( m = 2033.76 / 6179.319 ≈ 0.3291 \) kg. Hmm, still around 0.33 kg. Maybe that's correct? Wait, a bird's mass around 0.33 kg is reasonable (like a small bird). Wait, but let's check the calculation again. Wait, maybe the height is 62.99 m? Let's see, if h was 62.99 m, then gh = 9.8×62.99 = 617.302, then m = 2033.76 / 617.302 ≈ 3.295 kg, which is more reasonable for a bird. Wait, maybe the problem has a typo, but according to the given problem, h is 629.9 m. Wait, let's check the original problem again: "the tower is 629.9 m tall". So maybe it's correct. Wait, but let's proceed with the given numbers.

Wait, maybe I made a mistake in the formula. Wait, gravitational potential energy is \( PE = mgh \), so solving for m: \( m = PE/(gh) \). So with PE = 2033.76 J, g = 9.8 m/s², h = 629.9 m.

So \( m = 2033.76 / (9.8 * 629.9) \)

Calculate 9.8 629.9: 629.9 9.8. Let's do 629.9 10 = 6299, minus 629.9 0.2 = 125.98, so 6299 - 125.98 = 6173.02. Then 2033.76 / 6173.02 ≈ 0.3295 kg, which is approximately 0.33 kg.

Wait, but maybe the problem intended h to be 62.99 m? Let's check: if h = 62.99 m, then 9.8 * 62.99 = 617.302, 2033.76 / 617.302 ≈ 3.295 kg, which is about 3.3 kg, more reasonable for a bird. Maybe a typo in the problem, but we have to go with the given numbers.

But according to the problem, we have to use 629.9 m. So the calculation is as above.

Wait, let's do the division more precisely:

2033.76 ÷ 6173.02. Let's divide numerator and denominator by 2: 1016.88 ÷ 3086.51 ≈ 0.3295. So approximately 0.33 kg.

Answer:

The mass of the bird is approximately \(\boxed{0.33}\) kilograms (or more precisely, if we calculate 2033.76 / (9.8 * 629.9) = 2033.76 / 6173.02 ≈ 0.3295, which rounds to 0.33 kg).