A car can slow down at 5.10 m/s2 without skidding when coming to rest on a level road. What would its acceleration be if the road were inclined at 12o uphill? [Ask yourself – What is the same about the situation on the level road and on the incline?]
In order to determine the acceleration of the car on the inclined road, we need to consider the forces acting on the car. The key concept to remember here is that the acceleration of the car depends on the net force acting on it.
First, let's analyze the situation on the level road. Since the car is coming to rest without skidding, we know that the frictional force between the car's tires and the road is equal to the force that is decelerating the car. The frictional force can be calculated using the equation F_friction = µ * m * g, where µ is the coefficient of friction, m is the mass of the car, and g is the acceleration due to gravity.
On the level road, the frictional force is equal to m * a, where a is the acceleration. Therefore, we can write m * a = µ * m * g. We can cancel out the mass of the car from both sides of the equation, and we are left with a = µ * g.
Now, let's analyze the situation on the inclined road. The key difference here is that the force of gravity acting on the car is not acting vertically downward; it is acting at an angle, both downwards and parallel to the inclined road. The force parallel to the incline is what causes the car to accelerate or decelerate.
To find the acceleration on the inclined road, we can analyze the forces parallel to the incline. The force parallel to the incline can be calculated using the equation F_parallel = m * a_parallel, where a_parallel is the acceleration parallel to the incline.
The force parallel to the incline can be determined by resolving the weight of the car into its components. The component of the weight acting parallel to the incline can be calculated using the equation F_parallel = m * g * sin(θ), where θ is the angle of the incline.
Setting up the equation for the force parallel to the incline, we have m * a_parallel = m * g * sin(θ). Canceling out the mass of the car, we find a_parallel = g * sin(θ).
Therefore, the acceleration of the car on the inclined road would be a_parallel = g * sin(θ), where θ is the angle of the incline and g is the acceleration due to gravity.
In this case, the road is inclined at 12 degrees uphill. So, the acceleration of the car on the inclined road would be a_parallel = g * sin(12°).