You are driving your car over a circular-shaped bump in the road that has a radius of curvature of 75.9 m.
A)If the car is traveling at a constant speed of 18.3 m/s, calculate the apparent weight of your 56.1 kg passenger as you pass over the top of the bump.
B)What is the maximum speed that you can drive the car over the top of the bump without losing contact with the road?
A) Well, if you're driving over a circular-shaped bump, your passenger might feel like they're on a roller coaster! To calculate the apparent weight of your passenger as you pass over the top of the bump, we can use the formula:
Apparent weight = Normal force - Weight
Now, let's break it down step by step. First, let's calculate the normal force:
Normal force = mass of passenger x acceleration due to gravity
Normal force = 56.1 kg x 9.8 m/s²
Normal force = 548.58 N
Next, let's calculate the acceleration of your car as it goes through the bump. We can use the centripetal acceleration formula:
Centripetal acceleration = (velocity²) / radius of curvature
Centripetal acceleration = (18.3 m/s)² / 75.9 m
Centripetal acceleration = 4.448 m/s²
Now, using Newton's second law, we can find the apparent weight:
Apparent weight = Normal force - (mass of passenger x centripetal acceleration)
Apparent weight = 548.58 N - (56.1 kg x 4.448 m/s²)
Apparent weight = 548.58 N - 249.79 N
Apparent weight ≈ 298.79 N
So, the apparent weight of your passenger as you pass over the top of the bump is approximately 298.79 Newtons.
B) Now, let's figure out the maximum speed you can drive over the bump without losing contact with the road. In this case, we need to equate the centripetal force with the maximum friction force that can be exerted by the road on the tires.
Centripetal force = maximum friction force
The centripetal force can be given by the formula:
Centripetal force = (mass of car + mass of passenger) x centripetal acceleration
Since we want to find the maximum speed, we can assume that the maximum friction force will be equal to the static friction force. The static friction force can be given by:
Static friction force = coefficient of static friction x normal force
Now, let's put it all together. We can set up an equation:
(mass of car + mass of passenger) x centripetal acceleration = coefficient of static friction x normal force
(56.1 kg + mass of car) x 4.448 m/s² = coefficient of static friction x 548.58 N
Now, since we don't have the mass of the car, we can't directly solve for the maximum speed. We first need to know the coefficient of static friction between the tires and the road. Once we have that, we can plug it into the equation and solve for the maximum speed.
A) To calculate the apparent weight of the passenger, we need to determine the normal force acting on the passenger at the top of the bump.
At the top of the bump, the passenger is experiencing two forces: the gravitational force (Weight) and the normal force. The apparent weight is the sum of these two forces.
1. Calculate the gravitational force:
Weight = mass * acceleration due to gravity
Weight = 56.1 kg * 9.8 m/s^2
Weight = 549.78 N
2. Calculate the centripetal force:
Centripetal force = mass * velocity^2 / radius of curvature
Centripetal force = 56.1 kg * (18.3 m/s)^2 / 75.9 m
Centripetal force = 537.54 N
3. Calculate the normal force:
Apparent weight = Weight + Centripetal force
Apparent weight = 549.78 N + 537.54 N
Apparent weight = 1087.32 N
Therefore, the apparent weight of the passenger is 1087.32 N.
B) To determine the maximum speed without losing contact with the road, we need to calculate the maximum centripetal force that can be provided by the frictional force between the tires and the road.
The maximum centripetal force is given by:
Maximum centripetal force = frictional force
To calculate the frictional force, we can use the maximum static friction, which is determined by the coefficient of friction (μ) and the normal force.
1. Calculate the normal force:
Normal force = mass * acceleration due to gravity
Normal force = 56.1 kg * 9.8 m/s^2
Normal force = 549.78 N
2. Calculate the maximum static friction:
Maximum static friction = coefficient of friction * normal force
Assuming the coefficient of friction is 1 (typical for car tires on dry pavement):
Maximum static friction = 1 * 549.78 N
Maximum static friction = 549.78 N
3. Calculate the maximum centripetal force:
Maximum centripetal force = maximum static friction
Maximum centripetal force = 549.78 N
4. Calculate the maximum speed:
Maximum speed = √(maximum centripetal force * radius of curvature / mass)
Maximum speed = √(549.78 N * 75.9 m / 56.1 kg)
Maximum speed ≈ √(74076.66 N*m / 56.1 kg)
Maximum speed ≈ √(1320.10 m^2/s^2)
Maximum speed ≈ 36.34 m/s
Therefore, the maximum speed that you can drive the car over the top of the bump without losing contact with the road is approximately 36.34 m/s.
To calculate the apparent weight of the passenger and the maximum speed of the car over the bump, we need to use the concept of centripetal acceleration.
Centripetal acceleration is the acceleration directed towards the center of circular motion. In this case, it is the acceleration experienced by the car and the passenger as they go over the bump.
A) To calculate the apparent weight of the passenger, we need to consider the net force acting on the passenger when the car is at the top of the bump. The net force is the difference between the gravitational force and the centripetal force.
The gravitational force is given by the formula:
F_gravity = m * g
Where m is the mass of the passenger (56.1 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The centripetal force is given by the formula:
F_centripetal = m * a_centripetal
Where m is the mass of the passenger and a_centripetal is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
a_centripetal = v^2 / r
Where v is the speed of the car (18.3 m/s) and r is the radius of curvature of the bump (75.9 m).
Now, substituting the values into the equations:
F_gravity = m * g = 56.1 kg * 9.8 m/s^2 = 549.78 N
a_centripetal = (18.3 m/s)^2 / 75.9 m = 4.45 m/s^2
F_centripetal = m * a_centripetal = 56.1 kg * 4.45 m/s^2 = 249.45 N
The apparent weight of the passenger can be calculated as the net force acting on the passenger:
Apparent weight = F_gravity - F_centripetal = 549.78 N - 249.45 N = 300.33 N
Therefore, the apparent weight of the 56.1 kg passenger as the car passes over the top of the bump is 300.33 N.
B) To calculate the maximum speed of the car over the top of the bump, we need to determine the maximum centripetal acceleration that the car can handle without losing contact with the road.
At the maximum speed, the maximum centripetal acceleration is equal to the gravitational acceleration:
a_centripetal_max = g
Using the formula for centripetal acceleration:
a_centripetal_max = v_max^2 / r
Rearranging the formula, we can solve for the maximum speed:
v_max = sqrt(a_centripetal_max * r)
Substituting the values:
v_max = sqrt(9.8 m/s^2 * 75.9 m) = sqrt(753.42) m/s ≈ 27.45 m/s
Therefore, the maximum speed that the car can be driven over the top of the bump without losing contact with the road is approximately 27.45 m/s.