ou are the technical consultant for an action-adventure film in which a stunt calls for the hero to drop off a 18-m-tall building and land on the ground safely at a final vertical speed of 5 m/s. At the edge of the building's roof, there is a 100-kg drum that is wound with a sufficiently long rope (of negligible mass), has a radius of 0.4 m, and is free to rotate about its cylindrical axis with a moment of inertia
I0.
The script calls for the 69-kg stuntman to tie the rope around his waist and walk off the roof.
(a) Determine an expression for the stuntman's linear acceleration in terms of his mass m, the drum's radius r, and moment of inertia
I0.
(Use any variable or symbol stated above along with the following as necessary: g.)
a =
(b) Determine the required value of the stuntman's acceleration if he is to land safely at a speed of 5 m/s.
Use this value to calculate the moment of inertia of the drum about its axis.
(c) What is the angular acceleration of the drum?
(d) How many revolutions does the drum make during the fall?
(a) To determine the stuntman's linear acceleration, we need to analyze the forces acting on the system.
First, let's consider the gravitational force acting on the stuntman. The weight of the stuntman can be calculated using the equation:
Weight = mass * gravity
Since the mass of the stuntman is given as 69 kg, and the acceleration due to gravity is typically 9.8 m/s^2, we can calculate the weight as follows:
Weight = 69 kg * 9.8 m/s^2
The upward tension force in the rope can be calculated as the product of the moment of inertia of the drum (I0) and the angular acceleration (α), divided by the radius of the drum (r):
Tension = (I0 * α) / r
Since the rope is tied around the stuntman's waist, the tension force also equals the product of the stuntman's mass (m) and his linear acceleration (a):
Tension = m * a
Setting these two expressions for tension equal to each other, we have:
m * a = (I0 * α) / r
Rearranging the equation to solve for a, the acceleration, we get:
a = (I0 * α) / (r * m)
Therefore, the expression for the stuntman's linear acceleration in terms of his mass (m), the drum's radius (r), and the moment of inertia (I0) is:
a = (I0 * α) / (r * m)
(b) To determine the required value of the stuntman's linear acceleration for him to land safely at a speed of 5 m/s, we can simply substitute the given values into the equation we derived in part (a). Since the final vertical speed is 5 m/s, we can assume the initial speed is zero.
a = (I0 * α) / (r * m)
We know that the final vertical speed (vf) is given as 5 m/s, and the initial vertical speed (vi) is 0 m/s. We also know that the distance traveled (d) is 18 m (height of the building). We can use the kinematic formula:
vf^2 = vi^2 + 2ad
Plugging in the values, we have:
(5 m/s)^2 = (0 m/s)^2 + 2a * 18 m
Simplifying, we find:
25 m^2/s^2 = 36 a m/s^2
Now, we can solve for the required acceleration:
a = 25 m^2/s^2 / 36 m/s^2
Simplifying further:
a ≈ 0.694 m/s^2
This is the required value of the stuntman's acceleration.
To calculate the moment of inertia of the drum (I0), we can rearrange the equation in part (a) to solve for I0:
I0 = (a * r * m) / α
Substituting the known values:
I0 = (0.694 m/s^2 * 0.4 m * 69 kg) / α
(c) To determine the angular acceleration of the drum (α), we can analyze the torque acting on the drum. Since the drum is free to rotate, the torque generated by the tension force acting on the drum is equal to the moment of inertia times the angular acceleration:
Torque = I0 * α
From part (a), we know that the tension in the rope is equal to m * a:
Torque = Tension * r = (m * a) * r
Equating these two expressions, we have:
I0 * α = (m * a) * r
Therefore, the angular acceleration (α) of the drum is:
α = (m * a) / I0
(d) To calculate the number of revolutions the drum makes during the fall, we can use the relationship between the linear displacement and the angular displacement. Since the drum has a radius of 0.4 m, we can calculate the angular displacement (θ) using the formula:
θ = d / r
where d is the distance traveled (18 m).
θ = 18 m / 0.4 m
Simplifying further:
θ = 45 radians
Since there are 2π radians in one revolution, we can calculate the number of revolutions (N) as:
N = θ / (2π)
Substituting the value of θ, we have:
N = (45 radians) / (2π)