James Bond and the ever present bond girl are on a dirt bike escaping from a car load of bad guys. If we have the following conditions:
James Bond’s mass is 84 Kg
The Bond Girl is 48Kg
The Dirt Bike’s mass is 113.6 Kg
The tires on the dirt bike are 0.6m in diameter
With Bond and the girl on the dirt bike it can reach a top speed of 111 KpH
The coefficient of static friction between the bike and the road is 0.71
The coefficient of kinetic friction between the bike and the road is 0.4
a. How fast can the dirt bike go around a 90° curve, 6 m from the center of rotation, without losing traction?
b. As they pull out of the curve, the woman pulls a knife and tries to stab Bond. During the fight he knocks her off the bike. What is the maximum angular velocity of the tires, with just Bond on the bike?
a. To determine the speed at which the dirt bike can go around a 90° curve without losing traction, we need to calculate the maximum centripetal force that the tires can provide.
The centripetal force is given by the formula:
Centripetal Force = (Mass of the Bike + Mass of Bond + Mass of the Bond Girl) x (Velocity^2) / Radius
Using the given values:
Mass of the Bike = 113.6 kg
Mass of Bond = 84 kg
Mass of the Bond Girl = 48 kg
Velocity = 111 km/h = 111,000 m/3600 s ≈ 30.83 m/s
Radius = 6 m
Centripetal Force = (113.6 kg + 84 kg + 48 kg) x (30.83 m/s)^2 / 6 m
Now, we need to calculate the maximum friction force between the tires and the road to prevent the bike from losing traction.
Maximum Friction Force = Coefficient of Static Friction x Normal Force
The normal force can be calculated using the weight of the bike and occupants:
Normal Force = (Mass of the Bike + Mass of Bond + Mass of the Bond Girl) x Gravity
where Gravity = 9.8 m/s^2
Normal Force = (113.6 kg + 84 kg + 48 kg) x 9.8 m/s^2
Finally, we can calculate the maximum speed at which the dirt bike can go around the curve by equating the maximum friction force to the centripetal force:
Maximum Friction Force = Centripetal Force
Coefficient of Static Friction x Normal Force = (Mass of the Bike + Mass of Bond + Mass of the Bond Girl) x (Velocity^2) / Radius
0.71 x [(113.6 kg + 84 kg + 48 kg) x 9.8 m/s^2] = (113.6 kg + 84 kg + 48 kg) x (30.83 m/s)^2 / 6 m
Solving for Velocity, we get:
Velocity ≈ 20.89 m/s
Therefore, the dirt bike can go around a 90° curve, 6 m from the center of rotation, without losing traction at a speed of approximately 20.89 m/s.
b. When the Bond Girl is knocked off the bike, only the mass of Bond and the bike need to be considered.
To determine the maximum angular velocity of the tires with just Bond on the bike, we need to calculate the maximum centripetal force that the tires can provide.
Using only the mass of Bond and the bike:
Mass of Bond = 84 kg
Mass of the Bike = 113.6 kg
Velocity = 30.83 m/s
Using the same formula as before, we calculate the centripetal force:
Centripetal Force = (Mass of the Bike + Mass of Bond) x (Velocity^2) / Radius
Now, let's calculate the maximum friction force between the tires and the road:
Maximum Friction Force = Coefficient of Kinetic Friction x Normal Force
The normal force is the weight of Bond and the bike:
Normal Force = (Mass of the Bike + Mass of Bond) x Gravity
where Gravity = 9.8 m/s^2
Finally, we equate the maximum friction force to the centripetal force to calculate the maximum angular velocity of the tires:
Maximum Friction Force = Centripetal Force
Coefficient of Kinetic Friction x Normal Force = (Mass of the Bike + Mass of Bond) x (Velocity^2) / Radius
0.4 x [(113.6 kg + 84 kg) x 9.8 m/s^2] = (113.6 kg + 84 kg) x (30.83 m/s)^2 / 6 m
Solving for Velocity, we get:
Velocity ≈ 24.92 m/s
Therefore, the maximum angular velocity of the tires with just Bond on the bike is approximately 24.92 m/s.
To answer these questions, we need to consider the concept of centripetal acceleration and frictional forces. Here's how we can proceed:
a. To determine the maximum speed that the dirt bike can go around a 90° curve without losing traction, we need to calculate the maximum centripetal acceleration the bike can achieve without exceeding the static friction force.
The formula for centripetal acceleration is given by:
a = v^2 / r
Where:
a = centripetal acceleration
v = velocity
r = radius of curvature (6 m in this case)
To find the maximum velocity, we need to calculate the maximum centripetal acceleration first.
The maximum static friction force can be calculated using the formula:
F_static = μ_static * N
Where:
F_static = static friction force
μ_static = coefficient of static friction
N = normal force
The normal force can be calculated as:
N = (m_bond + m_girl + m_bike) * g
Where:
m_bond = mass of James Bond
m_girl = mass of the Bond Girl
m_bike = mass of the dirt bike
g = acceleration due to gravity (approximately 9.8 m/s^2)
Once we have the maximum static friction force, we can equate it to the centripetal force (which is given by the mass times the centripetal acceleration) to find the maximum velocity.
b. To calculate the maximum angular velocity of the tires with only Bond on the bike, we just need to consider the frictional force acting on the tires. The maximum angular velocity occurs when the kinetic friction force reaches its maximum value.
The maximum kinetic friction force can be calculated using the formula:
F_kinetic = μ_kinetic * N
Where:
F_kinetic = kinetic friction force
μ_kinetic = coefficient of kinetic friction
N = normal force (considering only Bond's mass in this case)
Once we have the maximum kinetic friction force, we can equate it to the centripetal force (which is given by the mass times the centripetal acceleration) to find the maximum angular velocity of the tires.
Now that we know the approach, we can proceed with the calculations. Let's start with question a.