After coming down a steep hill at a constant
speed of 49 m/s, a car travels along the circumference
of a vertical circle of radius 329 m
until it begins to climb another hill.
What is the magnitude of the resultant
force on the 170 kg driver of the car at the
lowest point on this circular path?
Answer in units of kN
To find the magnitude of the resultant force on the driver of the car at the lowest point on the circular path, we need to consider two main forces acting on the driver: the gravitational force and the centrifugal force.
The gravitational force acting on the driver can be calculated using the formula:
F_gravity = m * g
Where:
m = mass of the driver = 170 kg
g = acceleration due to gravity ≈ 9.8 m/s²
F_gravity = 170 kg * 9.8 m/s²
F_gravity = 1666 N
Now, let's calculate the centrifugal force acting on the driver. At the lowest point on the circular path, the driver is experiencing acceleration due to the circular motion. The centrifugal force can be calculated using the formula:
F_centrifugal = m * a
Where:
m = mass of the driver = 170 kg
a = acceleration due to circular motion
To find the acceleration, we need to first find the velocity of the car at the lowest point of the circular path. Since the car is traveling at a constant speed of 49 m/s and the radius of the circle is 329 m, we can use the formula for the velocity of an object moving in a circle:
v = (2 * pi * r) / T
Where:
v = velocity
r = radius
T = time period for one complete revolution
Since the car is at the lowest point on the circle, it takes half a revolution to reach that point. Therefore, the time period T is:
T = (1/2) * (2 * pi * r) / v
T = (1/2) * (2 * pi * 329 m) / 49 m/s
T ≈ 10.56 s
Now, we can calculate the acceleration a:
a = v² / r = (49 m/s)² / (329 m)
a ≈ 7.313 m/s²
Finally, we can calculate the centrifugal force:
F_centrifugal = m * a
F_centrifugal = 170 kg * 7.313 m/s²
F_centrifugal ≈ 1243 N
To find the magnitude of the resultant force on the driver at the lowest point, we need to consider both the gravitational force and the centrifugal force:
Resultant force = sqrt(F_gravity² + F_centrifugal²)
Resultant force = sqrt((1666 N)² + (1243 N)²)
Resultant force ≈ 2073 N
To convert the resultant force to kN (kilonewtons), divide by 1000:
Resultant force ≈ 2.073 kN
Therefore, the magnitude of the resultant force on the 170 kg driver at the lowest point on the circular path is approximately 2.073 kN.