A person doing a chin-up weighs 710 N, disregarding the weight of the arms. During the first 26.5 cm of the lift, each arm exerts an upward force of 356 N on the torso. If the upward movement starts from rest, what is the person's speed at this point?
To find the speed of the person at this point, we can use the principle of work and energy.
The work done on an object is equal to the change in its kinetic energy. In this case, the work done on the person can be calculated by multiplying the force exerted on the torso by the distance the torso moves upward.
Work = Force x Distance
Since the person starts from rest, the initial kinetic energy is zero. Therefore, the work done on the person will be equal to their final kinetic energy.
Now, let's calculate the work done on the person's torso:
Work = (Force exerted on torso) x (Distance moved by torso)
= 356 N x 0.265 m
= 94.34 J (Joules)
This work done on the torso is equal to the change in kinetic energy of the person.
Change in Kinetic Energy = Work
Since the initial kinetic energy is zero, the change in kinetic energy is equal to the final kinetic energy.
Final Kinetic Energy = 94.34 J
The formula for kinetic energy is:
Kinetic Energy = (1/2) x (mass) x (velocity)^2
Since the weight of the person is given in newtons, we can convert it to kilograms to use it in the formula.
Weight = mass x acceleration due to gravity
710 N = mass x 9.8 m/s^2
mass = 710 N / 9.8 m/s^2
mass ≈ 72.45 kg
Now we can rearrange the kinetic energy formula and solve for velocity:
Velocity = sqrt(2 x (Kinetic Energy) / (mass))
Velocity = sqrt(2 x 94.34 J / 72.45 kg)
Velocity ≈ 2.38 m/s
Therefore, the person's speed at this point is approximately 2.38 m/s.
To find the person's speed at the given point, we can apply the principle of conservation of mechanical energy. The initial potential energy is converted into kinetic energy at this point.
First, let's calculate the potential energy at the initial point:
Potential Energy (PE) = Weight * Height
Given that the weight of the person is 710 N and the height is 26.5 cm (which we convert to meters by dividing by 100):
PE = 710 N * (26.5 cm / 100) = 188.15 N*m
Since the upward movement starts from rest, there is no initial kinetic energy (KE₀ = 0).
The total mechanical energy at this point is the sum of the potential energy and kinetic energy at this point:
Total mechanical energy (E) = PE + KE
Since KE₀ = 0, the total mechanical energy at this point is equal to the potential energy:
E = 188.15 N*m
Now, let's equate the mechanical energy at this point to the kinetic energy at this point:
E = KE
0.5 * mass * velocity² = 188.15 N*m
Since mass is not given, we need to find it using the formula: weight = mass * gravity
Given that the weight is 710 N, and the acceleration due to gravity is approximately 9.8 m/s²:
710 N = mass * 9.8 m/s²
mass = 710 N / 9.8 m/s² = 72.45 kg
Now, substituting the mass value into the energy equation:
0.5 * 72.45 kg * velocity² = 188.15 N*m
Rearranging the equation to isolate velocity:
velocity² = (188.15 N*m) / (0.5 * 72.45 kg)
velocity² = 5.1668 m²/s²
Taking the square root of both sides to find the velocity:
velocity = √(5.1668 m²/s²)
velocity ≈ 2.27 m/s
Therefore, the person's speed at this point is approximately 2.27 m/s.