(#1.) A top starting from 40 rad/s takes 200 rad to stop. What is its rotational acceleration? (a constant)
(#2.) A stationary 100 Kg circular object (r= 0.20 m) is on a 35 degree incline. The coefficient of rolling friction between the block and the incline is 0.01. If it moves for 5 seconds, what is the angular displacement of the object at this time? (disregard torque)
For #1, I did: wf^2 = wi^2 + 2a(change in radians) and got a=4 radians/second^2. Is this right?
(I don't know how to do #2)
Yes on 1.
On 2, I don't know what a circular object is (sphere, cylinder, hoop). Each of those has a differing moment of inertial, and absorbs rolling friction differently.
The 2d drawing that goes with #2 shows a circle at the top of an incline. \
a circle could be a hoop,or a thin wheel.
Whichever one is the easiest to analyze.
To solve problem #1, you correctly used the equation wf^2 = wi^2 + 2aΔθ, where wf is the final angular velocity, wi is the initial angular velocity, a is the rotational acceleration, and Δθ is the change in radians.
Given that the initial angular velocity (wi) is 40 rad/s and the change in radians (Δθ) is 200 rad, you can plug in these values into the equation to solve for a:
wf^2 = wi^2 + 2aΔθ
0^2 = (40 rad/s)^2 + 2a(200 rad)
Simplifying the equation:
1600 rad^2/s^2 = 4000 rad^2 + 400a rad
400a rad = -2400 rad^2/s^2
Dividing both sides by 400 rad gives you the value of a:
a = -6 rad/s^2
Therefore, the rotational acceleration is -6 radians/second^2.
Now, moving on to problem #2, you are given the following information:
Mass (m) = 100 kg
Radius (r) = 0.20 m
Incline angle (θ) = 35 degrees
Coefficient of rolling friction (μ) = 0.01
Time (t) = 5 seconds
To find the angular displacement, you need to calculate the angular velocity (ω) and then multiply it by the time (t).
The formula for angular velocity is given by:
ω = v/r
Where v is the linear velocity. To find v, you can use the principles of forces and inclines.
The force of gravity acting on the circular object is given by:
Fg = m * g * sin(θ)
The force of rolling friction is:
Fr = μ * m * g * cos(θ)
The net force that causes the object to accelerate down the incline is the difference between the force of gravity and the force of rolling friction:
Net Force = Fg - Fr
Using Newton's second law, F = ma, you can set up the following equation:
Net Force = m * a
Substituting the net force equation:
m * a = m * g * sin(θ) - μ * m * g * cos(θ)
Simplifying:
a = g * (sin(θ) - μ * cos(θ))
Now, you can use this acceleration to find the final velocity (v) of the circular object at the bottom of the incline. The equation for linear motion with constant acceleration is:
v = u + at
Where u is the initial velocity, a is the acceleration, and t is the time.
In this case, the initial velocity (u) is 0 since the object starts from rest.
v = 0 + a * t
Substituting the acceleration equation:
v = (g * (sin(θ) - μ * cos(θ))) * t
Finally, you can find the angular velocity (ω) by dividing the linear velocity (v) by the radius (r):
ω = v / r
Substituting the value of v:
ω = ((g * (sin(θ) - μ * cos(θ))) * t) / r
Now that you have the angular velocity, you can find the angular displacement by multiplying it by the time (t):
Angular Displacement = ω * t
Substituting the value of ω:
Angular Displacement = ((g * (sin(θ) - μ * cos(θ))) * t) / r * t
Simplifying:
Angular Displacement = (g * (sin(θ) - μ * cos(θ)) * t^2) / r
Plug in the given values and calculate the angular displacement.