Centripital Problem: Solving an Old Streetcar Challenge

[cd80187]

Here is a homework problem that has really thrown me off.

An old streetcar rounds a flat corner of radius 9.4 m, at 14 km/h. What angle with the vertical will be made by the loosely hanging hand straps?

Clearly this has to do with centriptal force pushing out on the vehicle, but I really have no clue where to even start. If anyone could help me even to get started, that would be great. Clearly due to the lack of information, more than one euqation will be used to cancel out certain unknowns (such as mass), but I am not sure where to incorporate angles with centripital acceleration, which is the only thing you can really figure out from the information given.

 

[Doc Al]

It might be easier to imagine a small object hanging by a string tied to the ceiling of the streetcar. Find the angle the string makes by analyzing the forces acting on the object; you know it must be centripetally accelerating as it makes the turn.

 

[kyle8921]

I can understand how this problem may seem challenging at first. To solve this problem, we need to use the concept of centripetal force and its relationship with the velocity, radius, and mass of the object in circular motion.

First, let's identify the given information: the radius of the corner (9.4 m), the speed of the streetcar (14 km/h), and the direction of the hand straps (loosely hanging).

To start, we need to convert the speed from kilometers per hour to meters per second to match the units of the radius. We can do this by dividing 14 km/h by 3.6, which gives us a speed of 3.89 m/s.

Next, we can use the equation for centripetal force, Fc = m * v^2 / r, where m is the mass of the object, v is the velocity, and r is the radius. Since we do not have the mass of the streetcar, we can cancel it out by using the ratio of the centripetal force to the weight of the object, which is equal to the cosine of the angle between the two forces.

Therefore, we can rewrite the equation as Fc / mg = cosθ, where θ is the angle between the centripetal force and the weight of the streetcar. We can rearrange this equation to solve for the angle θ: θ = cos^-1 (Fc / mg).

Now, we need to find the centripetal force, which is equal to the mass of the streetcar multiplied by the centripetal acceleration (ac), which in this case is equal to v^2 / r. So, Fc = m * v^2 / r. We can plug in the values we have: Fc = m * (3.89 m/s)^2 / 9.4 m = 1.61 m * m/s^2 * m / kg = 1.61 N.

To find the weight of the streetcar (mg), we can use the equation mg = m * g, where g is the acceleration due to gravity (9.8 m/s^2). We can cancel out the mass again, and we are left with mg = 9.8 N.

Plugging these values into our equation for θ, we get θ = cos^-1 (1.61 N / 9.8 N