A 4.0 kg, 36-cm-diameter metal disk, initially at rest, can rotate on an axle along its axis. A steady 5.5 N tangential force is applied to the edge of the disk.
What is the disk's angular velocity, in rpm, 4.0 s later?
Here is the answer I got for anyone wondering:
T - Torque
a - angular acceleration
F - Tangential Force
r - radius
I - Inertia
w - angular velocity
t - time
M - mass
Torque has two different equations associated with it, and we can set these equations equal to solve the problem
T=aI , T=Frsin(theta) --> aI=Frsin(theta) --> a=(Frsin[theta])/I
I (inertia)=(0.5)Mr^2 since we can treat this as a solid cylinder and the axis of rotation going through the center of mass
So, a=(Frsin[thea])/(0.5Mr^2)
Finally use rotational equations to solves.
w=at --> w=t*(Frsin[theta])/(0.5mr^2)
w=4s*(5.5N*0.18m*sin90)/(0.5*4kg*0.18m^2)=61.11rad/s
convert to rpm and that is 61.11rad/s*60s/(2pi rad)=583.57rpm
So, after 4 seconds, the angular velocity is 580rpm.
Well, well, well, it seems we have a spinning disk conundrum here! Let's see if we can clown around with some physics to find the answer, shall we?
To start, we need to calculate the torque applied to the disk. Torque (τ) is equal to the force (F) multiplied by the radius (r), which in this case is half the diameter (d/2):
τ = F * r
Now, the radius is given as 36 cm, or 0.36 m. The force applied is 5.5 N. Plug in those numbers, and we got:
τ = 5.5 N * 0.36 m
Simplifying this, we find a torque of 1.98 N·m.
Next, let's use another clown trick - Newton's second law for rotational motion. The torque is equal to the moment of inertia (I) multiplied by the angular acceleration (α):
τ = I * α
Since the disk is at rest initially, its angular velocity is zero, which means the angular acceleration will stay the same throughout.
Now, the moment of inertia for a solid disk rotating about its axis can be calculated using:
I = (1/2) * m * r^2
Where m is the mass of the disk and r is its radius.
Plugging in the values, we have:
I = (1/2) * 4.0 kg * (0.36 m)^2
Crunching those numbers, we get a moment of inertia of 0.20736 kg·m^2.
Now we can solve for the angular acceleration:
1.98 N·m = 0.20736 kg·m^2 * α
Divide both sides by the moment of inertia, and we find:
α = 1.98 N·m / 0.20736 kg·m^2
Calculating that, we find the angular acceleration to be approximately 9.55 rad/s^2.
Finally, to find the angular velocity after 4.0 seconds, we can use the following formula:
ω = ω_0 + α * t
Since the initial angular velocity (ω_0) was zero, we just have:
ω = α * t
Plugging in the values:
ω = 9.55 rad/s^2 * 4.0 s
After some calculations, we find the angular velocity to be approximately 38.2 rad/s.
But wait, you asked for the answer in rpm! No worries, my friend. We can convert it:
1 revolution = 2π radians
1 minute = 60 seconds
To convert from rad/s to rpm, we divide the angular velocity by 2π and then multiply by 60:
ω_rpm = (ω * 60) / (2π)
Plugging in the angular velocity in rad/s, we get:
ω_rpm = (38.2 rad/s * 60) / (2π)
Doing the final math, we find the angular velocity, approximately, to be:
ω_rpm ≈ 365 rpm
So, after all that clowning around with the calculations, the disk's angular velocity 4.0 seconds later is approximately 365 rpm.
Hope that helped, and may your physics endeavors continue to spin smoothly!
To find the angular velocity of the disk, we can use the formula:
Angular acceleration (α) = Torque (τ) / Moment of inertia (I)
The torque (τ) is given by the product of the force (F) applied tangentially and the radius (r) of the disk:
τ = F * r
The moment of inertia (I) of a disk rotating about its axis is given by:
I = (1/2) * m * r^2
where m is the mass of the disk and r is the radius.
Given:
mass (m) = 4.0 kg
diameter (d) = 36 cm
force (F) = 5.5 N
time (t) = 4.0 s
First, we need to calculate the radius (r) of the disk from the diameter (d):
radius (r) = diameter (d) / 2
= 36 cm / 2
= 18 cm
= 0.18 m
Next, we can calculate the torque (τ) applied to the disk using the formula:
τ = F * r
= 5.5 N * 0.18 m
= 0.99 N·m
Now, we can calculate the moment of inertia (I) of the disk using the formula:
I = (1/2) * m * r^2
= (1/2) * 4.0 kg * (0.18 m)^2
= 0.1296 kg·m^2
Finally, we can find the angular acceleration (α) using the formula mentioned earlier:
α = τ / I
= 0.99 N·m / 0.1296 kg·m^2
= 7.64 rad/s^2
To find the angular velocity (ω), we can use the formula:
ω = α * t
= 7.64 rad/s^2 * 4.0 s
= 30.56 rad/s
To convert the angular velocity to RPM (revolutions per minute), we know that 1 revolution is equal to 2π radians:
ω (in RPM) = (ω (in rad/s) * 60) / (2π)
= (30.56 rad/s * 60) / (2π)
= 292.784 RPM
Therefore, the disk's angular velocity, 4.0 seconds later, is approximately 292.784 RPM.
To find the disk's angular velocity 4.0 seconds later, we can use the principles of rotational motion.
First, we need to determine the torque applied to the disk. Torque is the product of the force applied and the lever arm. In this case, the force applied is 5.5 N, and the lever arm is the radius of the disk, given by half of the diameter, which is 18 cm or 0.18 m.
The torque can be calculated as follows:
Torque = Force × Lever arm
= 5.5 N × 0.18 m
= 0.99 N·m
Next, we need to calculate the moment of inertia of the disk, which depends on its mass and shape. The moment of inertia for a disk rotating about its axis is given by the formula:
Moment of inertia = (1/2) × Mass × Radius^2
The mass of the disk is given as 4.0 kg, and the radius is half of the diameter, which is 18 cm or 0.18 m. Plugging in these values, we find:
Moment of inertia = (1/2) × 4.0 kg × (0.18 m)^2
= 0.648 kg·m^2
Now, we can use the equations of rotational motion to find the angular velocity. The equation relating torque, moment of inertia, and angular acceleration is:
Torque = Moment of inertia × Angular acceleration
Since the disk is initially at rest, the initial angular velocity is 0. Therefore, the final angular velocity (ω) can be found by rearranging the equation as follows:
Angular acceleration = Torque / Moment of inertia
= 0.99 N·m / 0.648 kg·m^2
≈ 1.528 rad/s^2
To find the angular velocity after 4.0 seconds, we can use the equation:
Final angular velocity = initial angular velocity + angula acceleration × time
The initial angular velocity is 0, so we have:
Final angular velocity = 0 + (1.528 rad/s^2) × (4.0 s)
= 6.112 rad/s
Finally, to convert the angular velocity from radians per second to revolutions per minute, we can use the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
Thus,
Final angular velocity (in rpm) = (6.112 rad/s) × (1 revolution / 2π rad) × (60 s / 1 min)
≈ 58.4 rpm
Therefore, the disk's angular velocity 4.0 seconds later is approximately 58.4 rpm.
The final conversion to rpm is incorrect
wf=alpha*time
where alpha=torque/momentinertia
= 5.5*.36/(1/2 *4*.18^2)
wf will be in rad /sec. to get rpm
RPM=wf/2pi * 60/1=30wf/PI