A meter stick is suspended vertically at a pivot point 27 cm from the top end. It is rotated on the pivot until it is horizontal and then released from rest. What will be its maximum angular velocity (in radians/second)?
potential energy at horizontal = m g h = m (9.81)(.50-.27) = 2.26 m Joules
kinetic energy at vertical = (1/2)I omega^2
I = m (1/12)(1^2) + m (.23)^2
= m (.136)
so
I omega^2 = .136 m omega^2
omega^2 = 2.26/.136
omega = 4.08 radians/second
To determine the maximum angular velocity of the meter stick, we can analyze the conservation of mechanical energy during its motion.
First, let's consider the initial potential energy of the meter stick when it is in a vertical position. The potential energy can be calculated as the product of the gravitational force and the height at the pivot point:
Potential energy (initial) = mgh
where m is the mass of the meter stick, g is the acceleration due to gravity, and h is the height at the pivot point.
Next, when the meter stick becomes fully horizontal, all of its initial potential energy will be converted into kinetic energy. The kinetic energy can be calculated as:
Kinetic energy (final) = (1/2) Iω^2
where I is the moment of inertia of the meter stick and ω is the angular velocity.
Since the meter stick is initially at rest and the final kinetic energy is equal to the initial potential energy, we can equate the expressions:
mgh = (1/2) Iω^2
We need to rearrange the equation to solve for the angular velocity ω:
ω = √((2mgh) / I)
To find the moment of inertia I, we need to know the physical dimensions and mass distribution of the meter stick. Assuming it is a uniform meter stick, its moment of inertia about an axis passing through its center of mass (at the pivot point) can be given by:
I = (1/3) mL^2
where L is the length of the meter stick.
Since we know the pivot point is 27 cm from the top end, and a meter stick is typically 100 cm long, we can substitute these values into the equation:
I = (1/3) m(100 cm)^2
Finally, we can substitute the expressions for I, m, g, and h into the equation for angular velocity:
ω = √((2mgh) / [(1/3) m(100 cm)^2])
Simplifying the equation gives:
ω = √((6gh) / (100 cm)^2)
Since we are looking for the angular velocity in radians per second, we should also convert the length from centimeters to meters:
100 cm = 1 meter
Now we can plug in the values and perform the calculations to find the maximum angular velocity of the meter stick.