A physics major is working to pay his college tuition by performing in a traveling carnival. He rides a motorcycle inside a hollow transparent plastic sphere. After gaining sufficient speed, he travels in a vertical circle with a radius of 13.4 . The physics major has a mass of 74.0 , and his motorcycle has a mass of 40.0 What minimum speed must he have at the top of the circle if the tires of the motorcycle are not to lose contact with the sphere?
m*v^2/R=mg centripetal force=the weight under the condition that traveled without losing contact with the sphere --> g= v^2/R--> v=(2gR)^(1/2)=(2*9.8*13.4)^(1/2)=16.206
To find the minimum speed required at the top of the circle, we need to consider the forces acting on the motorcycle at that point. At the top of the circle, there are two forces acting on the motorcycle: the gravitational force pulling it downward and the normal force pressing it against the sphere.
The normal force provides the centripetal force required to keep the motorcycle moving in a circle. The centripetal force is given by:
Fc = mv^2 / r
Where Fc is the centripetal force, m is the total mass (mass of the physics major plus the motorcycle), v is the speed, and r is the radius of the circle.
At the top of the circle, the normal force is equal to the sum of the gravitational force and the force due to the acceleration of the motorcycle. The gravitational force is given by:
Fg = mg
Where Fg is the gravitational force and g is the acceleration due to gravity.
To find the minimum speed, we can equate the two forces:
mv^2 / r = mg + ma
Simplifying the equation gives:
v^2 = rg + ra
Since the motorcycle and the physics major are moving together, their accelerations are the same. We can substitute a for v^2 / r:
v^2 = rg + rv^2 / r
Now, we can solve for v:
v^2 - rv^2 / r = rg
v^2 - v^2 = rg
v^2(1 - r) = rg
v^2 = rg / (1 - r)
Now, we substitute the given values:
v^2 = (9.8 m/s^2)(13.4 m) / (1 - 13.4 m)
v^2 = 130.12 m^2/s^2 / (-12.4)
v^2 ≈ -10.49 m^2/s^2
Since velocity cannot be negative, we take the positive square root:
v ≈ √(-10.49) m/s
However, this negative result indicates that it is impossible for the motorcycle to maintain contact with the sphere at the top of the circle. Therefore, there is no minimum speed for the motorcycle to avoid losing contact with the sphere.