A mass M of 5.80E-1 kg slides inside a hoop of radius R=1.50 m with negligible friction. When M is at the top, it has a speed of 5.15 m/s. Calculate the size of the force with which the M pushes on the hoop when M is at an angle of 41.0°.

To calculate the force with which the mass M pushes on the hoop when it is at an angle of 41.0°, we can apply the principles of circular motion. At any point on the circular path, the net force acting towards the center of the circle is provided by the centripetal force.

The centripetal force is given by the equation:

F = M * (V^2 / R)

Where:
F = Centripetal force
M = Mass of the object (in this case, mass M)
V = Velocity of the object (5.15 m/s)
R = Radius of the circular path (1.50 m)

Let's substitute the known values into the equation:

F = (5.80E-1 kg) * ((5.15 m/s)^2 / 1.50 m)

Simplifying the equation:

F = (5.80E-1 kg) * (26.5225 m^2/s^2 / 1.50 m)

F = (5.80E-1 kg) * (17.6817 N)

F ≈ 10.242 N

Therefore, the size of the force with which mass M pushes on the hoop when it is at an angle of 41.0° is approximately 10.242 N.

To calculate the force with which the mass M pushes on the hoop at an angle of 41.0°, you can use the principles of circular motion and centripetal force.

First, let's determine the velocity of the mass M at the angle of 41.0°. Since the system is frictionless, the conservation of mechanical energy can be applied. At a point at the top of the hoop, the gravitational potential energy is converted into kinetic energy:

mgh = (1/2)mv²

Where:
m = mass of the object (M)
g = acceleration due to gravity (9.8 m/s²)
h = height (in this case, equal to the diameter of the hoop, 2R)
v = velocity at the top

Let's solve for v:

v² = 2gh

Substituting the given values:
v² = 2 * 9.8 m/s² * 2 * (1.5 m)

v² = 58.8 m²/s²

Taking the square root of both sides:
v ≈ 7.67 m/s

Now, we can find the force with which M pushes on the hoop at the angle of 41.0°. In circular motion, the centripetal force, which points towards the center of the circle, is responsible for keeping an object moving in a curved path. It can be calculated using the formula:

F = m * ac

Where:
m = mass of the object (M)
ac = centripetal acceleration

The centripetal acceleration can be found using:

ac = v² / r

Where:
v = velocity of the object
r = radius of the hoop

Substituting the known values:
ac = (7.67 m/s)² / (1.5 m)

ac ≈ 39.82 m/s²

Finally, substitute the centripetal acceleration back into the centripetal force formula:

F = (5.80E-1 kg) * (39.82 m/s²)

Calculating the force:

F ≈ 2.30 N

Therefore, the size of the force with which the mass M pushes on the hoop at an angle of 41.0° is approximately 2.30 N.