A motorcycle has a constant speed of 24.4 m/s as it passes over the top of a hill whose radius of curvature is 135 m. The mass of the motorcycle and driver is 364 kg. Find the magnitude of (a) the centripetal force and (b) the normal force that acts on the cycle.

Why did the motorcycle go over the hill to begin with? Because it was tired of the constant flat road and wanted a change of scenery! Now let's find the answers, shall we?

(a) To find the magnitude of the centripetal force, we can use the formula:

Fc = (mv^2)/r

Where Fc is the centripetal force, m is the mass of the motorcycle and driver, v is the velocity, and r is the radius of curvature.

Plugging in the values:
Fc = (364 kg)(24.4 m/s)^2 / 135 m

Solve it and you'll have the answer!

(b) Now let's move on to the normal force. The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, it would be the force that keeps the motorcycle and driver from falling through the hill.

The normal force is equal to the gravitational force acting on the motorcycle and driver. So, we can calculate the gravitational force using the formula:

Fg = mg

Where Fg is the gravitational force, m is the mass, and g is the acceleration due to gravity.

In this case, Fg = (364 kg)(9.8 m/s^2)

That's your answer right there!

Now, let's get back to the motorcycle enjoying its ride. Keep on cruisin'!

To find the magnitude of the centripetal force and the normal force acting on the motorcycle, we can use the centripetal force equation and the concept of the normal force.

Given:
Speed of the motorcycle = 24.4 m/s
Radius of curvature of the hill = 135 m
Mass of the motorcycle and driver = 364 kg

Step 1: Find the centripetal force.
The centripetal force required to keep an object moving in a circular path is given by the formula:
Fc = (m × v^2)/r

Where:
Fc = centripetal force
m = mass of the object
v = velocity of the object
r = radius of curvature

Substituting the given values:
Fc = (364 kg × (24.4 m/s)^2) / 135 m

Calculating the centripetal force:
Fc = (364 kg × 596.36 m^2/s^2) / 135 m
Fc = 1600.825 m/s^2

Therefore, the magnitude of the centripetal force is approximately 1600.825 N.

Step 2: Find the normal force.
The normal force is the force exerted by a surface perpendicular to the object's motion, in this case, the hill.

Since the motorcycle is on a flat, horizontal surface at the top of the hill, the normal force is equal in magnitude and opposite in direction to the weight of the motorcycle and driver.

The weight of an object is given by:
Weight = mass × gravity

Where:
Weight = force due to gravity
mass = mass of the object
gravity = acceleration due to gravity (approximately 9.8 m/s^2)

Substituting the given value:
Weight = 364 kg × 9.8 m/s^2
Weight = 3575.2 N

Therefore, the magnitude of the normal force is approximately 3575.2 N.

So, the magnitude of the (a) centripetal force is approximately 1600.825 N, and the magnitude of the (b) normal force is approximately 3575.2 N.

To find the magnitude of the centripetal force acting on the motorcycle, we can use the formula:

Fc = m * v^2 / r

where Fc is the centripetal force, m is the mass of the motorcycle and driver, v is the velocity, and r is the radius of curvature.

Given:
m = 364 kg
v = 24.4 m/s
r = 135 m

Plugging in the values into the formula, we get:

Fc = (364 kg) * (24.4 m/s)^2 / 135 m

Calculating the expression on the right-hand side of the equation, we find:

Fc = 8032.72 N

Therefore, the magnitude of the centripetal force acting on the motorcycle is 8032.72 N.

To find the magnitude of the normal force acting on the motorcycle, we can use the fact that at the top of a hill, the normal force and gravitational force combine to provide the centripetal force.

In this case, we can equate the net force towards the center of the circle (provided by the normal force and gravitational force) to the centripetal force:

Fn + Fg = Fc

where Fn is the normal force and Fg is the gravitational force.

The gravitational force can be calculated using the formula:

Fg = m * g

where g is the acceleration due to gravity.

Given:
m = 364 kg
g = 9.8 m/s^2

Plugging in the values, we get:

Fn + (364 kg) * (9.8 m/s^2) = 8032.72 N

Simplifying the equation, we find:

Fn = 8032.72 N - (364 kg) * (9.8 m/s^2)

Calculating the expression on the right-hand side of the equation, we find:

Fn = 463.52 N

Therefore, the magnitude of the normal force acting on the motorcycle is 463.52 N.

(a) M V^2/R = 364*(24.4)^2/135

= 1605 newtons

(b) weight - (normal force) = centripetal force

normal force = Mg - 1605 = 3571 - 1605
= 1966 newtons
That is the vertical force exerted by the road on the motorcycle at the top of the hill.