A curve in a road forms part of a horizontal circle. As a car goes around it at constant speed 14.0 m/s, the total force exerted on the driver has magnitude 115 N. What are the magnitude and direction of the total vector force exerted on the driver if the speed is 25.0 m/s instead?
I thought you would use proportions, but its not right. THANK YOU
Ignoring gravity
v^2/R = centripetal acceleration
So force toward center = mv^2/R
115 N = m 14^2/R
so
m/R = 115/14^2
F = m 25^2/R = (115/14^2) (25^2)
You are welcome :)
In ∑F=m
r
v
2
, both m and r are unknown but remain constant.
Symbolically, write
∑F
slow
=(
r
m
)(14.0m/s)
2
and ∑F
fast
=(
r
m
)(18.0m/s)
2
Therefore, ∑F is proportional to v
2
and increases by a factor of (
14.0
18.0
)
2
as v increases from 14.0m/s to 18.0m/s. The total force at the higher speed is then
∑F
fast
=(
14.0
18.0
)
2
∑F
slow
=(
14.0
18.0
)
2
(130N)=215N
To understand the force exerted on the driver as the speed changes, we need to consider two concepts: centripetal force and inertia.
1. Centripetal Force:
When an object moves in a circular path, it experiences a force directed towards the center of the circle called the centripetal force. In this case, the centripetal force is provided by the friction between the car's tires and the road.
2. Inertia:
Inertia is the tendency of a moving object to continue moving in a straight line unless acted upon by an external force. As the car speeds up, the resistance to changing direction (inertia) will increase, requiring a larger centripetal force to keep the car moving in a circular path.
Now, let's solve the problem using the given information:
Given:
Speed of the car (v1) = 14.0 m/s
Total force on the driver (F1) = 115 N
We can calculate the magnitude of the centripetal force using the formula:
F1 = (mv1^2) / r
Where:
m = mass of the car
v1 = initial speed
r = radius of the circular path
Since the problem only asks about the change in magnitude and direction of the total force exerted on the driver, we don't need to calculate the precise values of m or r. We can assume that they remain constant.
Now, let's find the centripetal force for the initial speed:
115 N = (m * (14.0 m/s)^2) / r
Next, we need to calculate the centripetal force for the new speed:
Speed of the car (v2) = 25.0 m/s
We need to find the new force (F2). Using the same formula as before:
F2 = (m * (25.0 m/s)^2) / r
Now, let's compare the magnitudes and directions of force:
Since the mass (m) and radius (r) are assumed to remain the same, we can set up the following proportion to compare the forces:
F1 / F2 = (v1^2) / (v2^2)
Substituting the given values:
115 N / F2 = (14.0 m/s)^2 / (25.0 m/s)^2
Simplifying the equation:
115 N * (25.0 m/s)^2 = F2 * (14.0 m/s)^2
Solving for F2:
F2 = (115 N * (25.0 m/s)^2) / (14.0 m/s)^2
Calculating F2 will give you the new magnitude of the total vector force exerted on the driver.
Keep in mind that the direction will be the same as the initial force, as both forces are directed towards the center of the circular path in the same direction.
I hope this explanation helps! If you need further clarification, please let me know.