at the top of the loop , gravity is supplying the centripetal force
m * v^2 / r = m * g ... v^2 = r * g
0.5 m high.What is the minimum speed of the car at the top of
the loop for it to stay on the track?
m * v^2 / r = m * g ... v^2 = r * g
The equation goes like this: V = sqrt(R * g), where V is the minimum speed required, R is the radius of the loop (in this case, the radius of a vertical loop is equal to the height of the loop), and g is the acceleration due to gravity.
In this case, we have a loop that is 0.5 meters high, so that's the radius. And gravity, well, it's always 9.8 m/sĀ² on Earth.
Plugging in the numbers, we get V = sqrt(0.5 * 9.8). So, let me grab my calculator and do some quick math...
*dramatic calculator typing sounds*
The minimum speed for the car to stay on the track is approximately 3.13 m/s! So, make sure that little car is ready to go vroom vroom at that speed at the top of the loop, or else it might end up taking a detour to outer space!
At the top of the loop, the car has both kinetic energy (due to its motion) and potential energy (due to its height relative to the ground). The minimum speed occurs when the car just barely stays on the track.
At the top of the loop, all of the car's energy is in the form of potential energy. The potential energy at the top of the loop is given by the equation:
Potential energy = mass * gravity * height
The minimum speed can be calculated using the equation:
Potential energy = Kinetic energy
Therefore:
mass * gravity * height = (1/2) * mass * velocity^2
where:
mass = mass of the car
gravity = acceleration due to gravity (approximately 9.8 m/s^2)
height = height of the loop (0.5 m)
velocity = minimum velocity at the top of the loop
Simplifying the equation:
velocity^2 = 2 * gravity * height
Substituting the values:
velocity^2 = 2 * 9.8 m/s^2 * 0.5 m
velocity^2 = 9.8 m^2/s^2
Taking the square root of both sides:
velocity = ā(9.8 m^2/s^2)
velocity ā 3.13 m/s
Therefore, the minimum speed of the car at the top of the loop for it to stay on the track is approximately 3.13 m/s.