At 15 minutes past the hour, a 47.0 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 16.0 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.
At 15 minutes past the hour, a 47.0 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 16.0 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 find the magnitude of the torque exerted by the mouse on the minute-hand, we need to first understand the concept of torque and how it is calculated.
Torque, denoted as τ (tau), is the measure of the force that can cause an object to rotate around an axis. Mathematically, torque is defined as the product of the force applied perpendicular to the lever arm and the distance between the axis of rotation and the line of action of the force. It is given by the formula:
τ = F * r * sin(θ)
Where:
τ = torque
F = force applied
r = lever arm (distance between the axis of rotation and the line of action of the force)
θ = angle between the force and the lever arm
In this case, the mouse applies a force to the minute-hand, causing it to rotate. The torque exerted by the mouse is perpendicular to both the direction of force and the minute-hand, so θ is 90 degrees. We need to find the force applied by the mouse to calculate the torque.
To determine the force, we can use the equation:
Force = mass * acceleration
The mouse is sitting at the end of the minute-hand, which is rotating in a circular motion. The acceleration of an object moving in a circle can be calculated using the centripetal acceleration formula:
a = ω^2 * r
Where:
a = acceleration
ω = angular velocity (rate of change of angle over time)
r = radius of the circular path
The angular velocity (ω) can be found by dividing the angle covered in a given time by that time. Since it's given that the time is 15 minutes, we need to convert it to seconds:
15 minutes * (60 seconds/minute) = 900 seconds
The angle covered by the minute-hand in 15 minutes can be calculated using the formula:
Angle = (2π / 60) * t
Where:
Angle = angle covered
π = pi
t = time in seconds
Substituting the given values:
Angle = (2π / 60) * 900
Having found the angle, we can now calculate the angular velocity:
ω = Angle / t
Once we have the angular velocity, we can substitute it into the centripetal acceleration equation and solve for acceleration:
a = ω^2 * r
Now that we have the acceleration, we can calculate the force applied by the mouse using the equation:
Force = mass * acceleration
Finally, we can substitute the force, lever arm (16.0 cm = 0.16 m), and θ (90 degrees) into the torque equation:
τ = F * r * sin(θ)
This will give us the magnitude of the torque exerted by the mouse on the minute-hand.