An athlete at the gym holds a 3.0 kg steel ball in his hand. His arm is 62 cm long and has a mass of 3.8 kg, with the center of mass at 40% of the arm length from the shoulder.

What is the magnitude of the torque about his shoulder due to the weight of the ball and his arm if he holds his arm straight out to his side, parallel to the floor?

What is the magnitude of the torque about his shoulder due to the weight of the ball and his arm if he holds his arm straight, but 45∘ below horizontal?

To calculate the torque about the shoulder due to the weight of the ball and the arm, we need to consider the weight of both objects and the distance between their centers of mass and the shoulder.

1. Torque when the arm is held straight out to the side, parallel to the floor:

The distance between the shoulder and the center of mass of the arm is given as 40% of the arm length, which is 0.4 * 62 cm = 24.8 cm.

The torque due to the weight of the arm can be calculated as follows:
Torque_arm = (mass_arm) * (acceleration_due_to_gravity) * (distance)
= (3.8 kg) * (9.8 m/s^2) * (0.248 m)
≈ 9.4296 N*m

The torque due to the weight of the ball can be calculated as follows:
Torque_ball = (mass_ball) * (acceleration_due_to_gravity) * (distance)
= (3.0 kg) * (9.8 m/s^2) * (0.62 m)
≈ 18.156 N*m

Since both torques act in the same direction, we can simply add them to get the total torque:
Total torque = Torque_arm + Torque_ball
≈ 9.4296 N*m + 18.156 N*m
≈ 27.5856 N*m

Therefore, the magnitude of the torque about the shoulder is approximately 27.6 N*m.

2. Torque when the arm is held straight, but 45∘ below horizontal:

In this case, we need to find the new distance between the shoulder and the center of mass of the arm. This can be calculated as follows:

Distance_new = (arm length) * cos(angle)
= 0.62 m * cos(45∘)
= 0.62 m * 0.707 (approximating cos(45∘) as 0.707)
≈ 0.438 m

Using this new distance, we can calculate the torque due to the weight of the arm as follows:
Torque_arm_new = (mass_arm) * (acceleration_due_to_gravity) * (distance_new)
= (3.8 kg) * (9.8 m/s^2) * (0.438 m)
≈ 15.9192 N*m

The torque due to the weight of the ball remains the same as before:
Torque_ball = 18.156 N*m

Again, since both torques act in the same direction, we can simply add them to get the total torque:
Total torque = Torque_arm_new + Torque_ball
≈ 15.9192 N*m + 18.156 N*m
≈ 34.0752 N*m

Therefore, the magnitude of the torque about the shoulder is approximately 34.1 N*m when the arm is held straight, but 45∘ below horizontal.

To calculate the magnitude of the torque about the athlete's shoulder due to the weight of the ball and his arm, we need to know the length of the arm, the angle at which it is held, and the masses involved.

First, let's calculate the torque when the arm is held straight out to the side, parallel to the floor.

1. Calculate the torque due to the weight of the ball:
The torque due to the weight of the ball can be calculated using the formula: torque = weight * perpendicular distance from the pivot point.
Since the arm is held straight out to the side, the perpendicular distance from the shoulder to the ball can be calculated using the arm length and the 40% center of mass position.
perpendicular distance = arm length * (1 - center of mass position)
perpendicular distance = 0.62 m * (1 - 0.40)
perpendicular distance = 0.62 m * 0.60
perpendicular distance = 0.372 m

Now, calculate the torque due to the weight of the ball:
torque = mass * gravitational acceleration * perpendicular distance
torque = 3.0 kg * 9.8 m/s^2 * 0.372 m
torque ≈ 10.9064 Nm

2. Calculate the torque due to the weight of the arm:
The torque due to the weight of the arm can be calculated using the same formula, considering the arm's mass and center of mass position.
perpendicular distance = arm length * center of mass position
perpendicular distance = 0.62 m * 0.40
perpendicular distance = 0.248 m

Now, calculate the torque due to the weight of the arm:
torque = mass * gravitational acceleration * perpendicular distance
torque = 3.8 kg * 9.8 m/s^2 * 0.248 m
torque ≈ 9.4968 Nm

Finally, add up the torques from the ball and the arm to get the total torque about the shoulder:
total torque = torque of the ball + torque of the arm
total torque ≈ 10.9064 Nm + 9.4968 Nm
total torque ≈ 20.4032 Nm

Therefore, when the arm is held straight out to the side, parallel to the floor, the magnitude of the torque about the shoulder is approximately 20.4032 Nm.

Now let's calculate the magnitude of the torque when the arm is held straight, but 45∘ below horizontal.

1. We need to calculate the perpendicular distance from the shoulder to the line of action of the weight vector when the arm is at a 45∘ angle.
perpendicular distance = arm length * sin(angle)
perpendicular distance = 0.62 m * sin(45∘)
perpendicular distance ≈ 0.62 m * 0.7071
perpendicular distance ≈ 0.438 m

Now, calculate the torque due to the weight of the ball:
torque = mass * gravitational acceleration * perpendicular distance
torque = 3.0 kg * 9.8 m/s^2 * 0.438 m
torque ≈ 12.8804 Nm

2. Calculate the torque due to the weight of the arm:
perpendicular distance = arm length * center of mass position
perpendicular distance = 0.62 m * 0.40
perpendicular distance = 0.248 m

Now, calculate the torque due to the weight of the arm:
torque = mass * gravitational acceleration * perpendicular distance
torque = 3.8 kg * 9.8 m/s^2 * 0.248 m
torque ≈ 9.4968 Nm

Finally, add up the torques from the ball and the arm to get the total torque about the shoulder:
total torque = torque of the ball + torque of the arm
total torque ≈ 12.8804 Nm + 9.4968 Nm
total torque ≈ 22.3772 Nm

Therefore, when the arm is held straight, but 45∘ below horizontal, the magnitude of the torque about the shoulder is approximately 22.3772 Nm.

1st part ... forces are m*g , distances are given

2nd part ... same forces ... distances are reduced by geometry
... distances equal to 1st part multiplied by sin(45º)