How come a heavy truck and a small car can both travel safely at the same speed around an icy, banked-curve road? I would think that since mass is increased, normal force is therefore increased, and force of friction is therefore increased. And since the opposing force, the centripetal force, is just v^2/r, if both cars are going the same speed that force is the same. So wouldn't the forces of friction be different?
Similarly, if a driver wants to drive faster and puts some sand bags in his van aiming to increase the force of friction between tires and road, why will that not help? Again, I would think increased mass --> increased normal force --> increased force of friction. I understand that if he's driving faster, then the centripetal force also increases with v, but if he puts a bunch of sand bags in, the force of friction should increase, right?
forcefriction=centripetalforce
mu*mg= mv^2/r
notice mass divides out.
mu*g=v^2/r
mu= v^2/(rg)
Oh gotcha. But how come friction=centripetal force? I thought since net force is towards the center they can't cancel each other out like that...
While it may seem counterintuitive, a heavy truck and a small car can both travel safely at the same speed around an icy, banked-curve road due to the effects of friction and centripetal force.
When a vehicle travels around a curved road, there are two main forces at play: the force of friction and the centripetal force. The force of friction opposes the motion of the vehicle and is responsible for keeping the tires in contact with the road surface. The centripetal force is the force that pulls the vehicle inward toward the center of the curve.
Let's address the first scenario involving a heavy truck and a small car traveling at the same speed around an icy, banked-curve road. While it is true that an increase in mass would lead to an increase in the normal force (the force exerted by a surface to support the weight of an object), and subsequently an increase in the force of friction, the centripetal force is also dependent on the speed of the vehicle and the radius of the curve. At the same speed, the centripetal force required for both vehicles is the same, as you correctly mentioned (F_c = m * v^2 / r).
The force of friction, being dependent on the normal force, will indeed be higher for the heavier truck compared to the smaller car. However, the heavier truck also has a higher inertia due to its greater mass, which means it resists changes in motion more. This increased inertia counters the increased force of friction and keeps the truck stable around the curve, allowing it to travel safely at the same speed as the smaller car.
In the second scenario involving the driver wanting to drive faster by adding sandbags to the van, the situation is a bit different. While increasing the mass of the van by adding sandbags would indeed increase the normal force and subsequently the force of friction, it is important to note that the centripetal force required to make the van go around the curve at the increased speed also increases.
Since the force of friction is limited by the coefficient of friction and the normal force, there is a maximum amount of friction available to generate the centripetal force. As the van increases its speed, the required centripetal force increases as well. Eventually, the force of friction provided by the increased mass of the sandbags will not be enough to meet the increased centripetal force requirement, leading to a loss of traction and potentially causing the van to skid or lose control.
Therefore, while adding sandbags may increase the force of friction and stability at lower speeds, it becomes insufficient to counter the increased centripetal force at higher speeds. It's important to note that the maximum speed around a curve depends on various factors, including the coefficient of friction between the tires and the road surface, the shape and condition of the road, the vehicle's weight distribution, and the driver's skill and reaction time.
The relationship between mass, friction, and speed in the context of vehicles on icy, banked-curve roads can be a bit complex. Let's break down the factors involved and see why heavy trucks and small cars can both travel safely at the same speed in these conditions.
First, let's consider the forces acting on a vehicle as it negotiates a banked curve:
1. Centripetal Force: This force keeps the vehicle moving in a curved path and is directed towards the center of the curve. It depends on the mass of the vehicle and its speed (F = mv^2/r, where m is mass, v is speed, and r is the radius of the curve).
2. Force of Friction: This force acts in the opposite direction to the vehicle's motion and helps maintain traction between the tires and the road surface. It depends on the coefficient of friction (μ) and the normal force (N) exerted by the road on the vehicle. The frictional force is given by F_friction = μN.
Now, let's consider the scenarios you've mentioned:
1. Heavy Truck vs. Small Car: While it's true that the normal force (N) increases with increased mass, the force of friction (F_friction) also increases due to the higher normal force. Thus, the increased normal force cancels out the increased mass, resulting in a similar force of friction. As long as the centripetal force required to negotiate the curve is the same for both vehicles (same speed, same curve radius), the force of friction will be sufficient in both cases.
2. Sandbags in a Van: Placing sandbags in a van can increase its mass, thereby increasing the normal force (N). However, increasing the mass alone does not directly increase the force of friction. The force of friction depends on the coefficient of friction (μ) which remains constant between the tires and the road surface. Therefore, adding sandbags will not increase the force of friction. In fact, the extra weight may slightly decrease the overall handling and stability of the vehicle, especially if it is not distributed evenly.
In summary, when considering vehicles traveling at the same speed around an icy, banked-curve road, the forces at play - centripetal force and force of friction - remain appropriate for both heavy trucks and small cars due to the canceling effect of increased mass on normal force and the constant coefficient of friction. Additionally, adding sandbags to a van will not directly increase the force of friction, as it is dependent on the coefficient of friction rather than just mass.