You hold a uniform, 26.0-g pen horizontal with your thumb pushing down on one end and your index finger pushing upward 3.2 cm from your thumb. The pen is 13.9 cm long.
Find the two forces.
Force of Finger (Answer: 0.553N)
Force of Thumb (Answer: 0.299N)
Answers are provided above. Please show all work on how to get the answers.
13.9cm / 2 = 6.95cm
F(finger) 0.026kg * 9.8m/s^2 * 6.95cm / 3.2cm = 0.553N
F(thumb) 0.553N - 0.026kg * 9.8m/s^2 = 0.299N
To find the forces exerted by the thumb and the finger, we can use the principle of torque. The torque exerted by each force must be equal and opposite to maintain the equilibrium of the pen.
Here's how you can calculate the forces:
Step 1: Convert the mass of the pen into kilograms by dividing it by 1000:
Mass of the pen (m) = 26.0 g = 0.026 kg
Step 2: Calculate the distance from the thumb to the center of mass of the pen. Since the pen is horizontal, the center of mass is at the midpoint of its length:
Distance from the thumb to the center of mass (d1) = 13.9 cm / 2 = 6.95 cm = 0.0695 m
Step 3: Calculate the distance from the finger to the center of mass. This distance is given in the problem:
Distance from the finger to the center of mass (d2) = 3.2 cm = 0.032 m
Step 4: Calculate the torque exerted by the thumb and the finger:
Torque exerted by the thumb (τ1) = Force of the thumb (F1) × Distance from the thumb to the center of mass (d1)
Torque exerted by the finger (τ2) = Force of the finger (F2) × Distance from the finger to the center of mass (d2)
Since the forces exerted by the thumb and the finger must be equal and opposite to maintain equilibrium, we have:
τ1 = τ2
F1 × d1 = F2 × d2
Step 5: Solve the equation for one of the forces. Let's solve for the force of the finger (F2):
F2 = (F1 × d1) / d2
Given that F1 is the force of the thumb, we need to find F1 first.
Step 6: Calculate the torque exerted by the thumb (τ1):
τ1 = M × g × d1
Where M is the mass of the pen and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 7: Solve for F1 using the torque equation:
F1 = τ1 / d1
Step 8: Substitute the values into the equations and calculate the forces:
F1 = τ1 / d1 = (M × g × d1) / d1
F2 = (F1 × d1) / d2
Substitute the given values:
M = 0.026 kg
g = 9.8 m/s^2
d1 = 0.0695 m
d2 = 0.032 m
Calculate F1:
F1 = (0.026 kg × 9.8 m/s^2 × 0.0695 m) / 0.0695 m = 0.199 N
Calculate F2:
F2 = (0.199 N × 0.0695 m) / 0.032 m = 0.428 N
Round the final answers to the appropriate number of significant figures:
Force of the finger (F2) = 0.428 N ≈ 0.43 N
Force of the thumb (F1) = 0.199 N ≈ 0.20 N
Therefore, the forces exerted by the finger and the thumb are approximately 0.43 N and 0.20 N, respectively.