Answer:
10.9 J.
Step-by-step explanation:
We can begin by finding the gravitational potential energy (GPE) of the mass at its initial position, where the string makes an angle θ of 51.5º with the vertical:
GPE = mgh
where m is the mass, g is the acceleration due to gravity, and h is the vertical height above the ground. In this case, h can be expressed as L(1 - cos θ), where L is the length of the string:
h = L(1 - cos θ) = 1.2 m (1 - cos 51.5º) ≈ 0.743 m
Using g = 9.81 m/s^2 and m = 2.98 kg, we can calculate the initial GPE:
GPE = mgh = 2.98 kg × 9.81 m/s^2 × 0.743 m ≈ 21.9 J
As the mass falls, its potential energy is converted into kinetic energy (KE). At the bottom of its swing, all of the GPE is converted into KE, so we can use the conservation of energy principle to find the KE at this point:
GPE = KE
21.9 J = (1/2)mv^2
where v is the velocity of the mass at the bottom of its swing. Solving for v, we get:
v = sqrt(2GPE/m) = sqrt(2(21.9 J)/2.98 kg) ≈ 2.59 m/s
The work done by gravity can be found using the work-energy principle:
Work by gravity = KE - GPE
At the bottom of the swing, the GPE is zero, so the work done by gravity is just the KE:
Work by gravity = (1/2)mv^2 = (1/2)(2.98 kg)(2.59 m/s)^2 ≈ 10.9 J
Therefore, the work done by gravity by the time the string is in a vertical position for the first time is approximately 10.9 J.