Experiment in this week (week 8) is to study rotational dynamics. Consider a stainless steel annular disk with an outer radius 63 mm and inner radius 7.4 mm. The mass of the disk is 1340 grams. Keep your answers to at least 4 significant digits.
(a) What is the moment of inertia of the stainless steel annular disk?
.00266 kg m2
(b) The stainless steel annular disk is allowed to rotate on a frictionless table with the rotation axis at its center. The disk has a small cylinder rigidly mounted at the top concentrically. The cylinder's radius is 12.5 mm, and the mass of the cylinder is negligible. A string is wrapped around the cylinder, and a hanging mass of 20.8 g is tied at the other end of the string. When the mass falls under gravity, it causes the stainless steel annular disk to rotate. Ignoring the string's mass, and assuming that the string's motion is frictionless, what is the angular acceleration of the stainless steel annular disk?
rad/s2
(c) What is the angular speed of the stainless steel annular disk 4.1 seconds after the hanging mass is released from rest?
rad/s
(d) At what speed is the hanging mass falling at this time?
m/s
(e) What is the kinetic energy of the falling mass at this time?
J
(f) What is the rotational kinetic energy of the stainless steel annular disk at this time?
J
(g) How much distance has the hanging mass been falling by this time?
m
I've gotten the first two but others i'm just lost can someone help?
You know a) and b)
c) speed=angularacceleration*time
d)tangential speed=angularspeed*radius
e) 1/2 m v^2
f) 1/2 I w^2
g) distance=1/2 angularspeed*time^2
Sure! Let's go through each question step by step.
(b) To find the angular acceleration of the stainless steel annular disk, we can use Newton's second law for rotation. The torque applied to the disk is equal to the moment of inertia times the angular acceleration.
The torque applied to the disk is equal to the tension in the string multiplied by the radius of the cylinder. The tension in the string is equal to the weight of the hanging mass (20.8 g) multiplied by the acceleration due to gravity (9.8 m/s^2).
So, we have:
Torque = Tension * Radius = (0.0208 kg) * (9.8 m/s^2) * (0.0125 m)
Next, we can calculate the angular acceleration using the formula:
Torque = Moment of inertia * Angular acceleration
Angular acceleration = Torque / Moment of inertia
Plugging in the values we have so far, we get:
Angular acceleration = [(0.0208 kg) * (9.8 m/s^2) * (0.0125 m)] / (0.00266 kg m^2)
Calculating the result will give you the answer in rad/s^2.
(c) To find the angular speed of the stainless steel annular disk after 4.1 seconds, we can use the kinematic equation for rotational motion:
Angular speed = Initial angular speed + (Angular acceleration * Time)
Since the hanging mass is released from rest, the initial angular speed is 0 rad/s. We already know the angular acceleration (from part b) and the time is given as 4.1 seconds. Plugging in these values, you can find the angular speed in rad/s.
(d) The speed at which the hanging mass is falling can be determined using the equation:
Speed = Acceleration * Time
In this case, the acceleration due to gravity is the acceleration. You can plug in the value of the acceleration due to gravity (9.8 m/s^2) and the time (4.1 seconds) to find the speed in m/s.
(e) The kinetic energy of the falling mass can be calculated using the equation:
Kinetic energy = (1/2) * Mass * Speed^2
Here, you can use the mass of the hanging mass (20.8 g) and the speed (calculated in part (d)) to find the kinetic energy in Joules.
(f) The rotational kinetic energy of the stainless steel annular disk at this time can be found using the formula:
Rotational kinetic energy = (1/2) * Moment of inertia * Angular speed^2
We already have the moment of inertia (from part (a)) and the angular speed (from part (c)). Plugging in these values will give you the rotational kinetic energy in Joules.
(g) To find the distance the hanging mass has fallen, we can use the equation for distance traveled under constant acceleration:
Distance = (1/2) * Acceleration * Time^2
Here, the acceleration is the acceleration due to gravity (9.8 m/s^2) and the time is given as 4.1 seconds. Plugging in the values will give you the distance in meters.
I hope this helps! Let me know if you have any further questions.