Question

A skateboarder is attempting to make a circular arc of radius r = 17 m in a parking lot. The total mass of the skateboard and skateboarder is m = 91 kg. The coefficient of static friction between the surface of the parking lot and the wheels of the skateboard is μs = 0.59.

Part (a) What is the maximum speed, in meters per second, he can travel through the arc without slipping?
vm =?
Part (b) He speeds up very slightly and begins to slide. The coefficient of kinetic friction is μk = 0.13. What is the new magnitude, in meters per squared second, of his radial acceleration?
ac =_m/s squared

Answer 1

In part (a), the maximum speed the skateboarder can travel through the arc without slipping is 7.82 m/s. In part (b), the new magnitude of his radial acceleration when he begins to slide is 1.274 m/s².

Step-by-step explanation:

Part (a): To find the maximum speed at which the skateboarder can travel through the arc without slipping, we need to equate the maximum force of static friction with the centripetal force. The maximum force of static friction can be calculated using the formula:

fsMax = μs * m * g

where fsMax is the maximum force of static friction, μs is the coefficient of static friction, m is the total mass, and g is the acceleration due to gravity.

The centripetal force can be calculated using the formula:

Fc = m * ac

where Fc is the centripetal force and ac is the radial acceleration.

Equating the two forces, we have:

fsMax = Fc

μs * m * g = m * ac

Simplifying, we find:

ac = μs * g

To find the maximum speed, we can use the formula: vm = sqrt(r * ac)

Substituting the given values:

vm = sqrt(17 * 0.59 * 9.8) = 7.82 m/s

Therefore, the maximum speed he can travel through the arc without slipping is 7.82 m/s.

Part (b): To find the new magnitude of his radial acceleration when he begins to slide, we can use the formula:

ac = μk * g

Substituting the given values:

ac = 0.13 * 9.8 = 1.274 m/s²

Therefore, the new magnitude of his radial acceleration is 1.274 m/s².