A 0.630 kg wooden block is spun around on a wooden table. The wooden block is being spun around on a 2.00 m long massless wire. The wooden block has air being pushed out behind it causing a thrust force of 3.30 N. The air is being pushed out at 70.0° from the radial line as shown in the figure. If the block is initially at rest and the coefficient of friction is 0.370, what is the tension in the wire after 16.0 revolutions?
uoit question
70 degrees? what direction?
The radial line is in line with the string on the diagram and is going out from the block. The angle is in between the radial line and the block.
90i
To find the tension in the wire after 16.0 revolutions, we need to analyze the forces acting on the wooden block.
First, let's calculate the gravitational force acting on the wooden block:
Gravitational force (Fg) = mass (m) × acceleration due to gravity (g)
Given that the mass of the wooden block is 0.630 kg and the acceleration due to gravity is approximately 9.8 m/s²:
Fg = 0.630 kg × 9.8 m/s² = 6.174 N
Next, let's consider the normal force (N) acting on the block. The normal force is equal in magnitude and opposite in direction to the gravitational force when the block is on a horizontal surface.
Normal force (N) = Gravitational force (Fg) = 6.174 N
Now, let's find the frictional force (Ff) acting on the block. The frictional force can be calculated using the frictional force equation:
Frictional force (Ff) = coefficient of friction (μ) × normal force (N)
Given that the coefficient of friction is 0.370:
Ff = 0.370 × 6.174 N = 2.282 N
Since the block is initially at rest, the thrust force (Ft) provided by the air pushes the block forward and accelerates it. The net force (Fnet) acting on the block is given by the difference between the thrust force and the frictional force:
Net force (Fnet) = Thrust force (Ft) - Frictional force (Ff)
Fnet = 3.30 N - 2.282 N = 1.018 N
The centripetal force (Fc) required to keep the block moving in a circular path is provided by the tension in the wire.
Centripetal force (Fc) = tension in the wire
The centripetal force can be calculated using the following equation:
Fc = mass (m) × radial acceleration (ar)
Given that the block is rotating at a constant speed, its radial acceleration is given by the equation:
Radial acceleration (ar) = v²/r
Where v is the linear velocity and r is the radius of the circular path.
The linear velocity (v) can be calculated using the formula:
v = 2πr/T
Where T is the time taken to complete one revolution.
Given that the length of the wire is 2.00 m, the radius of the circular path is 2.00 m as well.
One revolution is equal to 2π radians, so the time taken for 16.0 revolutions (T) can be calculated as follows:
Time (T) = (time for one revolution) × (number of revolutions)
Since the block is rotating at a constant speed, the time for one revolution (T1) is the same as the time for 16.0 revolutions (T):
T = T1
Finally, we can calculate the tension in the wire (Fc) using the net force (Fnet) and the radial acceleration (ar):
Fc = mass (m) × radial acceleration (ar) = Fnet
Let's compute the tension in the wire after 16.0 revolutions: