A poundal is the force required to accelerate a mass of 1 lbm at a rate of 1 ft/s2, and a slug is the mass of an object that will accelerate at a rate of 1 ft/s2 when subjected to a force of 1 lbf.
(a) Calculate the mass is slugs and the weight in poundals of a 175 lbm man (i) on earth and (ii) on the moon, where the acceleration of gravity is one-sixth of its value on earth.
(b) A force of 355 poundals is exerted on a 25.0-slug object. At what rate (m/s2) does the object accelerate?
The Correct Answer and Explanation is :
Let’s break down each part of the problem.
Part (a)
Given:
- 1 lbm = mass unit in the imperial system.
- 1 lbf = force unit in the imperial system.
- 1 poundal (pdl) = force required to accelerate 1 lbm at 1 ft/s².
We are tasked with calculating the mass in slugs and weight in poundals for a 175 lbm man both on Earth and on the Moon.
Step 1: Mass in slugs
- The relationship between lbm (pound-mass) and slug is:
( 1 \, \text{slug} = 32.174 \, \text{lbm} ). - The mass of the man in slugs:
[
\text{Mass in slugs} = \frac{\text{Mass in lbm}}{32.174}
]
[
\text{Mass in slugs} = \frac{175 \, \text{lbm}}{32.174} \approx 5.43 \, \text{slugs}
]
Step 2: Weight in poundals
- The weight in poundals is the force exerted by the gravitational field on the mass.
- On Earth, the acceleration due to gravity (( g_{\text{earth}} )) is approximately ( 32.174 \, \text{ft/s}^2 ).
- Weight in poundals is calculated as:
[
W_{\text{earth}} = \text{Mass in lbm} \times g_{\text{earth}} = 175 \times 32.174 \, \text{pdl}
]
[
W_{\text{earth}} = 5630.45 \, \text{pdl}
] - On the Moon, the acceleration due to gravity is ( \frac{1}{6} ) of Earth’s gravity, so ( g_{\text{moon}} = \frac{32.174}{6} \approx 5.362 \, \text{ft/s}^2 ).
- Weight on the Moon in poundals:
[
W_{\text{moon}} = 175 \times 5.362 \, \text{pdl}
]
[
W_{\text{moon}} \approx 937.85 \, \text{pdl}
]
So the man’s mass in slugs is approximately ( 5.43 \, \text{slugs} ), and his weight is:
- 5630.45 poundals on Earth,
- 937.85 poundals on the Moon.
(ii) Final Results for (a):
- Mass on Earth (slugs): 5.43 slugs.
- Weight on Earth (poundals): 5630.45 poundals.
- Weight on the Moon (poundals): 937.85 poundals.
Part (b)
We are given:
- Force = 355 poundals
- Mass = 25 slugs
We need to calculate the acceleration.
Step 1: Use Newton’s second law of motion
Newton’s second law states that:
[
F = m \times a
]
Where:
- ( F ) is the force (355 poundals),
- ( m ) is the mass (25 slugs),
- ( a ) is the acceleration (in ft/s²).
To find the acceleration:
[
a = \frac{F}{m}
]
Substitute the values:
[
a = \frac{355 \, \text{poundals}}{25 \, \text{slugs}}
]
Since 1 poundal = 0.031081 \, N, and 1 slug = 14.5939 kg, we need to convert both the force and mass to SI units. So, we can first calculate the acceleration in ft/s² and then convert it to m/s².
[
355 \, \text{poundals} = 355 \times 0.031081 \, \text{N} \approx 11.05 \, \text{N}
]
[
25 \, \text{slugs} = 25 \times 14.5939 \, \text{kg} \approx 364.85 \, \text{kg}
]
Now, calculate the acceleration in m/s²:
[
a = \frac{11.05 \, \text{N}}{364.85 \, \text{kg}} \approx 0.0303 \, \text{m/s}^2
]
Final Answer:
The object will accelerate at a rate of approximately 0.0303 m/s².
Explanation:
In part (a), we calculated the mass in slugs and weight in poundals of a 175 lbm man both on Earth and on the Moon. The man’s mass remains constant at 5.43 slugs, but his weight varies depending on the gravitational acceleration at each location. On Earth, where gravity is stronger, his weight is about 5630.45 poundals, and on the Moon, it is much less (937.85 poundals) because gravity is weaker there.
In part (b), we used Newton’s second law to determine the acceleration of an object subjected to a force. We first converted the force from poundals to newtons and the mass from slugs to kilograms to use the SI unit system for consistency. After applying the equation ( F = m \times a ), we found the acceleration to be about 0.0303 m/s². This highlights how the combination of force and mass determines the acceleration of an object in motion.