An athlete at the gym holds a 5.54kg steel ball in his hand. His arm is 80.4cm long and has a mass of 4.26kg. What is the magnitude of the torque about his shoulder if he holds his arm straight out to his side, parallel to the floor?
b).What is the magnitude of the torque about his shoulder if he holds his arm straight, but 58.2deg below horizontal?
To find the torque about the athlete's shoulder in each scenario, we need to use the formula:
Torque = (Force) x (Distance) x (sinθ)
Where:
- Force is the force exerted by the steel ball.
- Distance is the distance from the shoulder to the point of application of the force.
- θ is the angle between the force and the lever arm.
Given information:
- Mass of the steel ball (m) = 5.54 kg
- Length of athlete's arm (L) = 80.4 cm
- Mass of the athlete's arm (M) = 4.26 kg
a). When the arm is straight out to the side, parallel to the floor, the angle θ between the force and the lever arm is 90 degrees.
1. Convert the length of the athlete's arm to meters:
Length of arm (L) = 80.4 cm = 0.804 m
2. Calculate the distance from the shoulder to the point of application of the force:
Distance (d) = 0.804 m
3. Calculate the force exerted by the steel ball:
Force (F) = m x g (where g is the acceleration due to gravity)
Force (F) = 5.54 kg x 9.8 m/s^2
4. Calculate the torque about the shoulder:
Torque = F x d x sinθ
θ = 90 degrees
Torque = (5.54 kg x 9.8 m/s^2) x 0.804 m x sin(90 degrees)
b). When the arm is straight, but tilted 58.2 degrees below the horizontal, the angle θ between the force and the lever arm is 58.2 degrees.
1. Calculate the distance from the shoulder to the point of application of the force (lever arm):
Distance (d) = 0.804 m
2. Calculate the force exerted by the steel ball:
Force (F) = m x g (where g is the acceleration due to gravity)
Force (F) = 5.54 kg x 9.8 m/s^2
3. Calculate the torque about the shoulder:
Torque = F x d x sinθ
θ = 58.2 degrees
Torque = (5.54 kg x 9.8 m/s^2) x 0.804 m x sin(58.2 degrees)
Please note that the numeric values in the above equations are rounded to two decimal places.
To calculate the torque about the shoulder, we need to multiply the force applied (in this case, the weight of the steel ball) by the perpendicular distance between the point of rotation (shoulder) and the line of action of the force. The equation for torque is given by:
Torque = Force × Distance
a) When the athlete holds his arm straight out to his side, parallel to the floor, the force exerted by the steel ball is its weight. The weight of an object can be calculated using the equation:
Weight = Mass × Acceleration due to gravity
Given:
Mass of the steel ball (m) = 5.54 kg
Acceleration due to gravity (g) = 9.8 m/s²
Using the given values, we can calculate the weight of the steel ball:
Weight = Mass × Acceleration due to gravity
Weight = 5.54 kg × 9.8 m/s²
Once we have the weight of the steel ball, we can calculate the torque by multiplying it by the distance between the shoulder and the line of action of the force. In this case, the distance is the length of the athlete's arm.
Length of the arm (L) = 80.4 cm = 0.804 m
To calculate the torque, we use the equation:
Torque = Weight × Length of the arm
b) When the athlete holds his arm straight, but at an angle of 58.2 degrees below horizontal, the perpendicular distance between the line of action of the force and the shoulder needs to be calculated.
To find the perpendicular distance, we can use trigonometry. The perpendicular distance (d) can be calculated using the equation:
d = Length of the arm × sin(angle)
Using the given angle and length of the arm, we can calculate the perpendicular distance (d).
Once we have the perpendicular distance, we can calculate the torque using the equation:
Torque = Weight × Perpendicular distance
By plugging in the respective values, we can find the magnitude of the torque in both cases.