A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, as shown in the figure (see the figure ). The pulley is a uniform disk with mass 10.4 and radius 51.0 and turns on frictionless bearings. You measure that the stone travels a distance 12.4 during a time interval of 2.80 starting from rest.
h=a•t²/2
a = 2•h/t² =2•12.4/2.8²=3.16 m/s².
According to the Newton’s Second Law for rotation
the torque is the moment of inertia times
an angular acceleration
M=I•ε,
M = T•R.
I=m•R²/2.
ε =a/R.
T•R = (m•R²/2)• (a/R)=
= m•R• a/2.
T= m•a/2.
The tension is
T=10.4•3.16/2=16.4 N.
Forces on the stone are
ma = mg – T,
m = T/(g-a) =16.4/(9.8-3.16)=2.47 kg.
Well, that sounds like quite a pulleying situation! I hope you're not taking me for granite with this question. Let's get the ball rolling and find some answers!
To figure this out, we'll need to keep a few things in mind. The first thing to note is that the pulley is a uniform disk with a mass of 10.4 kg and a radius of 51.0 m. The stone travels a distance of 12.4 m during a time interval of 2.80 s, starting from rest.
Now, let's calculate the acceleration of the stone. We can use the equation:
acceleration = (final velocity - initial velocity) / time
Since the stone starts from rest, the initial velocity is 0 m/s. The final velocity can be calculated using the equation:
final velocity = (distance traveled) / time
Plugging in the values:
final velocity = 12.4 m / 2.80 s
Now we have the final velocity. But remember, the stone is attached to the pulley, so the pulley will also have an angular velocity. The angular velocity of the pulley can be calculated using:
angular velocity = final velocity / radius
Plugging in the values:
angular velocity = final velocity / 51.0 m
With the angular velocity of the pulley, we can calculate its rotational kinetic energy using the formula:
rotational kinetic energy = 0.5 * (moment of inertia) * (angular velocity)^2
The moment of inertia of a disk can be calculated using the formula:
moment of inertia = 0.5 * (mass of disk) * (radius of disk)^2
Plugging in the values:
moment of inertia = 0.5 * 10.4 kg * (51.0 m)^2
Finally, we can calculate the rotational kinetic energy:
rotational kinetic energy = 0.5 * (moment of inertia) * (angular velocity)^2
Now, that should give you a good idea about the motion of the stone and the pulley. Just remember, even though the situation is pulley complicated, with a little bit of math, we can solve it!
To find the tension in the wire, we can use Newton's second law for rotational motion. The torque exerted on the pulley is equal to the moment of inertia of the pulley times its angular acceleration.
Step 1: Find the moment of inertia of the pulley.
The moment of inertia of a solid disk rotating about its central axis is given by the formula: I = (1/2) * m * r^2, where m is the mass of the disk and r is its radius.
Given:
Mass of the pulley, m = 10.4 kg
Radius of the pulley, r = 51.0 cm = 0.51 m
Substituting the values into the formula:
I = (1/2) * 10.4 kg * (0.51 m)^2
I = 1.32 kg * m^2
Step 2: Find the angular acceleration of the pulley.
The angular displacement of the pulley can be calculated using the formula: θ = ω0 * t + (1/2) * α * t^2, where θ is the angular displacement, ω0 is the initial angular velocity (zero in this case), t is the time interval, and α is the angular acceleration.
Given:
θ = 12.4 m
t = 2.80 s
ω0 = 0
Rearranging the formula, we get:
α = (2θ) / t^2
Substituting the values into the formula:
α = (2 * 12.4 m) / (2.80 s)^2
α = 1.77 rad/s^2
Step 3: Calculate the torque exerted on the pulley.
The torque exerted on the pulley is given by the formula: τ = I * α, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
Given:
I = 1.32 kg * m^2
α = 1.77 rad/s^2
Substituting the values into the formula:
τ = 1.32 kg * m^2 * 1.77 rad/s^2
τ = 2.3292 N * m
Step 4: Find the tension in the wire.
The tension in the wire can be calculated using the formula: T = (1/2) * (m * g - τ / r), where T is the tension, m is the mass of the stone, g is the acceleration due to gravity, τ is the torque on the pulley, and r is the radius of the pulley.
Given:
T = ?
m = ?
g = 9.8 m/s^2
τ = 2.3292 N * m
r = 0.51 m
Substituting the values into the formula and solving for T:
T = (1/2) * (m * g - τ / r)
T = (1/2) * (m * 9.8 m/s^2 - 2.3292 N * m / 0.51 m)
T = (4.9 m * g - 4.5573 N)
T = 4.9 m * 9.8 m/s^2 - 4.5573 N
T = 48.02 m - 4.5573 N
T = 48.02 m - 4.5573 N
Unfortunately, we don't have enough information to calculate the tension in the wire without knowing the mass of the stone. Please provide the mass of the stone for further assistance.
To solve this problem, we can use the concept of rotational kinematics and equations of motion.
First, let's define the given information:
- Mass of the pulley (m) = 10.4 kg
- Radius of the pulley (r) = 51.0 cm = 0.51 m
- Distance traveled by the stone (d) = 12.4 m
- Time interval (t) = 2.80 s
- Initial angular velocity of the pulley (ω_initial) = 0 rad/s (starting from rest)
Now, we need to find the final angular velocity of the pulley (ω_final). We can use the equation of motion for rotational motion:
θ = ω_initial * t + (1/2) * α * t^2
Here, θ represents the angle covered by the pulley, α is the angular acceleration, ω_initial is the initial angular velocity, and t is the time interval.
Since the pulley starts from rest (ω_initial = 0), the equation simplifies to:
θ = (1/2) * α * t^2
In this case, the angle covered by the pulley is the arc length traveled by the stone on the pulley's outer rim, which is equal to the distance traveled by the stone (θ = d).
Therefore, the equation becomes:
d = (1/2) * α * t^2
Solving for α, we get:
α = (2 * d) / (t^2)
Now, we can find the final angular velocity (ω_final) using another equation of motion for rotational motion:
ω_final = ω_initial + α * t
Since ω_initial is 0, the equation simplifies to:
ω_final = α * t
Now we can substitute the value of α from the previous equation:
ω_final = (2 * d) / (t^2) * t
Simplifying further:
ω_final = (2 * d) / t
Finally, we can substitute the given values into the equation to find the final angular velocity:
ω_final = (2 * 12.4 m) / (2.80 s)
Calculating:
ω_final = 8.857 rad/s
Therefore, the final angular velocity of the pulley is 8.857 rad/s.