A thin, light wire is wrapped around the rim of a wheel, as shown in the following figure. The wheel rotates without friction about a stationary horizontal axis that passes through the center of the wheel. The wheel is a uniform disk with radius 0.400 m. An object of mass 4.50 kg is suspended from the free end of the wire. The system is released from rest and the suspended object descends with constant acceleration. If the suspended object moves downward a distance of 5.15 m in 2.15 s, what is the mass of the wheel?
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To solve this problem, we can use the concept of rotational dynamics and equate the system's torque to the net torque applied by gravity.
Step 1: Determine the acceleration of the suspended object.
Given:
Distance (s) = 5.15 m
Time (t) = 2.15 s
We can use the formula for constant acceleration:
s = ut + (1/2)at^2
Since the object is released from rest (u = 0), the formula simplifies to:
s = (1/2)at^2
Rearranging the equation to solve for acceleration (a):
a = (2s) / t^2
= (2 * 5.15 m) / (2.15 s)^2
a ≈ 4.43 m/s^2
Step 2: Calculate the net torque.
Given:
Radius of the wheel (r) = 0.400 m
Mass of the object (m) = 4.50 kg
Acceleration of the object (a) = 4.43 m/s^2
The net torque acting on the wheel is the torque exerted by gravity on the suspended object. It can be calculated using the equation:
τ_net = m * g * r
where:
m is the mass of the object
g is the acceleration due to gravity (approximated as 9.8 m/s^2)
r is the radius of the wheel
τ_net = (4.50 kg) * (9.8 m/s^2) * (0.400 m)
τ_net ≈ 17.64 N.m
Step 3: Calculate the moment of inertia (I) of the wheel.
The moment of inertia of a uniform disk can be calculated using the formula:
I = (1/2) * m * r^2
where:
m is the mass of the wheel
We want to determine the mass of the wheel, so we re-arrange the equation to solve for m:
m = (2 * I) / r^2
Given:
Radius of the wheel (r) = 0.400 m
We can substitute this value into the equation:
m = (2 * I) / (0.400 m)^2
m = (2 * I) / 0.1600 m^2
Step 4: Substitute the values into the equation.
Given:
Net torque (τ_net) ≈ 17.64 N.m
Radius of the wheel (r) = 0.400 m
Using the equation from step 3:
m = (2 * τ_net) / r^2
= (2 * 17.64 N.m) / (0.400 m)^2
Calculating the value:
m ≈ 276.38 kg
Therefore, the mass of the wheel is approximately 276.38 kg.
To find the mass of the wheel, we can use Newton's second law of motion, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
In this case, the net force acting on the system is the tension in the wire (T). The tension in the wire is responsible for both the acceleration of the hanging object and the rotation of the wheel.
First, let's calculate the acceleration of the hanging object. We can use the equation of motion for an object with constant acceleration:
d = v_initial * t + (1/2) * a * t^2
where
d = distance traveled (5.15 m),
v_initial = initial velocity (0 m/s, as it is released from rest),
t = time (2.15 s),
a = acceleration (which is the same for both the hanging object and the wheel).
Substituting the given values into the equation, we can solve for the acceleration:
5.15 m = 0 * 2.15 s + (1/2) * a * (2.15 s)^2
5.15 m = (1/2) * a * (2.15 s)^2
a = (2 * 5.15 m) / (2.15 s)^2
a ≈ 2.94 m/s^2
Now, we can calculate the net force on the system using the equation F = ma:
F = (4.50 kg) * (2.94 m/s^2)
F ≈ 13.23 N
Since the tension in the wire is responsible for the acceleration of both the hanging object and the rotation of the wheel, we can consider the tension as the net force.
The tension in the wire can be found by considering the forces acting on the hanging object. There are two forces acting on the object: its weight (mg) acting downwards and the tension (T) acting upwards. Since the object is accelerating downwards, the net force is given by:
F = mg - T
T = mg - F
T = (4.50 kg) * (9.8 m/s^2) - 13.23 N
T ≈ 40.05 N
The tension in the wire is 40.05 N. Since the wire is wrapped around the rim of the wheel, this tension also provides the torque (rotational force) to rotate the wheel.
The torque can be calculated using the equation:
T = I * α
where
T = torque,
I = moment of inertia of the wheel, and
α = angular acceleration of the wheel.
In this case, since the wheel is a uniform disk, the moment of inertia (I) is given by:
I = (1/2) * m * r^2
where
m = mass of the wheel (what we are trying to find), and
r = radius of the wheel (0.400 m).
Substituting the given values into the equation, we can solve for the mass of the wheel:
40.05 N = (1/2) * m * (0.400 m)^2
40.05 N = (1/2) * m * 0.160 m^2
40.05 N = 0.08 m * m
m ≈ 40.05 N / 0.08 m
m ≈ 500.63 kg
Therefore, the mass of the wheel is approximately 500.63 kg.