Mass M1 has twice the mass of M2. M1 and M2 are raised from a height of 5 m to 25 m, and from 14 m to 24 m, respectively. What is the ratio of work done on mass M1 to that done on mass M2?
A. 1 : 4
B. 25 : 12
C. 4 : 1
D. 2 : 1
E. 5 : 1
Let M1 = 2kg. M2 = 1 kg.
W1 = F1*d1 = (M1g)*(25-5) = (2*9.8)*20 =
W2 = (1*9.8)*(24-14) =
W1/W2 =
To find the ratio of work done on mass M1 to that done on mass M2, we can use the formula for work done:
Work = force x distance
We know that the distance for both masses is the same (25 m - 5 m = 20 m for M1 and 24 m - 14 m = 10 m for M2). Let's assume the force applied to M1 is F1, and the force applied to M2 is F2.
From Newton's second law (F = ma), we can say that the force applied to an object is equal to the mass of the object multiplied by the acceleration:
F = m*a
In this case, M1 has twice the mass of M2, so we can write:
F1 = 2m*a1
F2 = m*a2
Since the distance is the same for both masses, and work is equal to force multiplied by distance, we can compare the work done on M1 to that done on M2 by comparing their respective forces:
Work on M1 / Work on M2 = (F1 x distance) / (F2 x distance)
The distances cancel out, leaving us with:
Work on M1 / Work on M2 = F1 / F2
= (2m*a1) / (m*a2)
= 2a1 / a2
Now we need to find the ratio of accelerations. Acceleration is determined by the change in velocity over time, so we can write:
a1 = (vf1 - vi1) / t1
a2 = (vf2 - vi2) / t2
Where vf is the final velocity, vi is the initial velocity, and t is the time. In this case, the initial and final velocities are the same for both masses (vf1 = 25 m/s, vi1 = 5 m/s, vf2 = 24 m/s, vi2 = 14 m/s).
Plugging these values into the equations for acceleration:
a1 = (25 m/s - 5 m/s) / t1
a2 = (24 m/s - 14 m/s) / t2
Since the time is not given, it cancels out in the ratio of accelerations:
a1 / a2 = (25 m/s - 5 m/s) / (24 m/s - 14 m/s)
= 20 m/s / 10 m/s
= 2
Finally, plugging this ratio of accelerations back into the equation for the ratio of work:
Work on M1 / Work on M2 = 2a1 / a2
= 2(2) / 1
= 4 / 1
Therefore, the ratio of work done on mass M1 to that done on mass M2 is:
4 : 1
The correct answer is C. 4 : 1.
To find the ratio of work done on mass M1 to that done on mass M2, we need to calculate the work done on each mass.
The work done on an object is given by the formula:
Work = force × distance
Where force is the upward force exerted on the object and distance is the vertical displacement of the object.
In this case, both masses are raised vertically, so the force exerted on each mass will be equal to its weight, and the distance will be the difference in height between the initial and final positions.
Let's calculate the work done on M1 first:
Force on M1 = mass of M1 × acceleration due to gravity
= 2M2 × g
Distance for M1 = final height of M1 - initial height of M1
= 25 m - 5 m
= 20 m
Work done on M1 = Force on M1 × Distance for M1
= 2M2 × g × 20 m
Now let's calculate the work done on M2:
Force on M2 = mass of M2 × acceleration due to gravity
= M2 × g
Distance for M2 = final height of M2 - initial height of M2
= 24 m - 14 m
= 10 m
Work done on M2 = Force on M2 × Distance for M2
= M2 × g × 10 m
To find the ratio of work done on M1 to that done on M2, we divide the work done on M1 by the work done on M2:
Ratio = (2M2 × g × 20 m) / (M2 × g × 10 m)
= 2 × 20 / 10
= 40 / 10
= 4
Therefore, the ratio of work done on mass M1 to that done on mass M2 is 4 : 1, which corresponds to option C. 4:1.