Sure.
Can I use your answer to part-'a' ?
If the angular acceleration is actually 32 rev/min², than
after 1.2 min, it has reached the speed of
(32 rev/min²) x (1.2 min) = 38.4 rev/min .
Check:
If the initial speed is zero and the final speed is 38.4 rpm,
then the average speed during the acceleration period is
(1/2) (0 + 38.4) = 19.2 rpm average
At an average speed of 19.2 rpm for 1.2 min,
it covers
(19.2 rev/min) x (1.2 min) = 23.04 revs .
That's pretty close to the "23" in the question, so I think that
everything here is in order.
Answer:
The best option is for the following option m = 15 [g] and V = 5 [cm³]
Explanation:
We have that the density of a body is defined as the ratio of mass to volume.
![Ro =m/V](https://tex.z-dn.net/?f=Ro%20%3Dm%2FV)
where:
Ro = density = 3 [g/cm³]
Now we must determine the densities with each of the given values.
<u>For m = 7 [g] and V = 2.3 [cm³]</u>
![Ro=7/2.3\\Ro=3.04 [g/cm^{3} ]](https://tex.z-dn.net/?f=Ro%3D7%2F2.3%5C%5CRo%3D3.04%20%5Bg%2Fcm%5E%7B3%7D%20%5D)
<u>For m = 10 [g] and V = 7 [cm³]</u>
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<u>For m = 15 [g] and V = 5 [cm³]</u>
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<u>For m = 21 [g] and V = 8 [cm³]</u>
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Answer:
A. 4,9 m/s2
B. 2,0 m/s2
C. 120 N
Explanation:
In the image, 1 is going to represent the monkey and 2 is going to be the package. Let a_mín be the minimum acceleration that the monkey should have in the upward direction, so the package is barely lifted. Apply Newton’s second law of motion:
![\sum F_y=m_1*a_m_i_n = T-m_1*g](https://tex.z-dn.net/?f=%5Csum%20F_y%3Dm_1%2Aa_m_i_n%20%3D%20T-m_1%2Ag)
If the package is barely lifted, that means that T=m_2*g; then:
![\sum F_y =m_1*a_m_i_n=m_2*g-m_1*g](https://tex.z-dn.net/?f=%5Csum%20F_y%20%3Dm_1%2Aa_m_i_n%3Dm_2%2Ag-m_1%2Ag)
Solving the equation for a_mín, we have:
![a_m_i_n=((m_2-m_1)/m_1)*g = ((15kg-10kg)/10kg)*9,8 m/s^2 =4,9 m/s^2](https://tex.z-dn.net/?f=a_m_i_n%3D%28%28m_2-m_1%29%2Fm_1%29%2Ag%20%3D%20%28%2815kg-10kg%29%2F10kg%29%2A9%2C8%20m%2Fs%5E2%20%3D4%2C9%20m%2Fs%5E2)
Once the monkey stops its climb and holds onto the rope, we set the equation of Newton’s second law as it follows:
For the monkey: ![\sum F_y = m_1*a \rightarrow T-m_1*g=m_1*a](https://tex.z-dn.net/?f=%5Csum%20F_y%20%3D%20m_1%2Aa%20%5Crightarrow%20T-m_1%2Ag%3Dm_1%2Aa)
For the package: ![\sum F_y = m_2*a \rightarrow m_2*g - T = m_2*a](https://tex.z-dn.net/?f=%5Csum%20F_y%20%3D%20m_2%2Aa%20%5Crightarrow%20m_2%2Ag%20-%20T%20%3D%20m_2%2Aa)
The acceleration a is the same for both monkey and package, but have opposite directions, this means that when the monkey accelerates upwards, the package does it downwards and vice versa. Therefore, the acceleration a on the equation for the package is negative; however, if we invert the signs on the sum of forces, it has the same effect. To be clearer:
For the package: ![\sum F_y = -m_2*a \rightarrow T-m2*g=-m_2*a \rightarrow m_2*g -T=m_2 *a](https://tex.z-dn.net/?f=%5Csum%20F_y%20%3D%20-m_2%2Aa%20%5Crightarrow%20T-m2%2Ag%3D-m_2%2Aa%20%5Crightarrow%20m_2%2Ag%20-T%3Dm_2%20%2Aa)
We have two unknowns and two equations, so we can proceed. We can match both tensions and have:
![m_1*a+m_1*g=m_2*g-m_2*a](https://tex.z-dn.net/?f=m_1%2Aa%2Bm_1%2Ag%3Dm_2%2Ag-m_2%2Aa)
Solving a, we have
![(m_1+m_2)*a =(m_2 - m1)*g\\\\a=((m_2-m_1)/(m_1+m_2))*g \rightarrow a=((15kg-10kg)/(10kg+15kg))*9,8 m/s^2\\\\a= 2,0 m/s^2](https://tex.z-dn.net/?f=%28m_1%2Bm_2%29%2Aa%20%3D%28m_2%20-%20m1%29%2Ag%5C%5C%5C%5Ca%3D%28%28m_2-m_1%29%2F%28m_1%2Bm_2%29%29%2Ag%20%5Crightarrow%20a%3D%28%2815kg-10kg%29%2F%2810kg%2B15kg%29%29%2A9%2C8%20m%2Fs%5E2%5C%5C%5C%5Ca%3D%202%2C0%20m%2Fs%5E2)
We can then replace this value of a in one for the sums of force and find the tension T:
![T = m_1*a+m_1*g \rightarrow T=m_1*(a+g)\\\\T = 10kg*(2,0 m/s^2+9,8 m/s^2) \\\\T = 120 N](https://tex.z-dn.net/?f=T%20%3D%20m_1%2Aa%2Bm_1%2Ag%20%5Crightarrow%20T%3Dm_1%2A%28a%2Bg%29%5C%5C%5C%5CT%20%3D%2010kg%2A%282%2C0%20m%2Fs%5E2%2B9%2C8%20m%2Fs%5E2%29%20%5C%5C%5C%5CT%20%3D%20120%20N)
From the average speed you can fix an equation:
Average speed = distance / time
You know the average speed = 65.1 kg / h, then
65.1 = distance / total time,
where total time is the time traveling plus 22.0 minutes
Call t the time treavelling and pass 22 minutes to hours:
65.1 = distance / [t + 22/60] ==> distance = [t + 22/60]*65.1
From the constant speed, you can fix a second equation
Constant speed = distance / time traveling
94.5 = distance / t ==> distance = 94.5 * t
The distance is the same in both equations, then you have:
[t +22/60] * 65.1 = 94.5 t
Now you can solve for t.
65.1t + 22*65.1/60 = 94.5t
94.5t - 65.1t = 22*65.1/60
29.4t = 23.87
t = 23.87 / 29.4
t = 0.812 hours
distance = 94.5 km/h * 0.812 h = 76.7 km
Answers: 1) 0.81 hours, 2) 76.7 km