At 15 minutes past the hour, a 50 g mouse runs up a grandfather clock and sits himself down at the end of the minute-hand. If the minute-hand for the clock is 16 cm long, what is the magnitude of the torque exerted by the mouse on the minute-hand? Give your answer in the standard units of N m
To calculate the torque exerted by the mouse on the minute-hand, we first need to understand the concept of torque and its formula.
Torque is the measure of the force that can cause an object to rotate around an axis. It depends on the force applied and the distance from the axis. The formula for torque is:
Torque = Force × Distance × sin(θ)
- Force: The force applied perpendicular to the lever arm. In this case, it is the weight of the mouse, which can be calculated using the formula weight = mass × acceleration due to gravity (w = m × g).
- Distance: The distance of the point of application of force from the axis of rotation. In this case, it is the length of the minute-hand.
- θ: The angle between the force vector and the lever arm vector. In this case, it is 90 degrees since the force and the lever arm are perpendicular.
Now let's calculate the torque:
1. Calculate the force:
Mass of the mouse = 50 g = 0.05 kg (convert grams to kilograms)
Acceleration due to gravity, g = 9.8 m/s²
Force = mass × acceleration due to gravity = 0.05 kg × 9.8 m/s²
2. Calculate the distance:
Length of the minute-hand = 16 cm = 0.16 m (convert centimeters to meters)
3. Calculate the torque:
Torque = Force × Distance × sin(θ)
θ = 90 degrees
Torque = Force × Distance × sin(90°)
Since sin(90°) = 1, we can simplify the equation to:
Torque = Force × Distance
Substituting the values we calculated earlier:
Torque = (0.05 kg × 9.8 m/s²) × 0.16 m
Now, multiply and solve for torque:
Torque = 0.0784 N m
Therefore, the magnitude of the torque exerted by the mouse on the minute-hand is 0.0784 N m.