Mech 3: Sphere Sliding Problem - Mass, Radius, and Acceleration Calculations

[Elphaba]

Mech 3. Sphere. PLEASE help...

A particle of mass m slides down a fixed, frictionless shpere of radius R, starting from rest at the top.

(a) in terms of m, g, R, and (theta), determine each of the following for the particle while it is sliding on the sphere.

i. the Kinetic energy of the particle
ii. the centipetal acceleration of the mass (would you still use Ac=v^2/r?)
iii. The tangential acceleration of the mass (What's this?)

(b) determine the value of (Theta) at which the particle leaves the sphere.

 

[dextercioby]

HINTS:

1.Make a drawing.
2.Mark all forces that are present between the three bodies from the system
3.Make the symplifying assmption that the sphere's mass is infinite,so the sphere will not move,slide,roll...
4.Write down Newton's second law of dynamics in vector form for the sliding body down the sphere.
5.Chose a system of axis whiwh will enable u to project the eq.written at 4. in a symple form.
6.Write down the condition for the sliding body to lose contat with the sphere.
7.After finding the angle at 6. all kinetic variables (quantities) will be known:liner and angular acceleration;linear and angular velocity and you'll easily find the kinetic energy of the particle up until the moment of "breakin' loose'.

Daniel.

 

[wais]

(a) i. The kinetic energy of the particle can be calculated using the formula KE = 1/2 * m * v^2, where m is the mass of the particle and v is its velocity. In this case, the particle starts from rest, so its initial velocity is 0. Therefore, the kinetic energy of the particle is 0.

ii. The centripetal acceleration of the mass can still be calculated using the formula Ac = v^2/r, where v is the velocity of the particle and r is the radius of the sphere. However, in this case, the velocity of the particle is constantly changing as it slides down the sphere. Therefore, we need to use the formula v = sqrt(2*g*h), where g is the acceleration due to gravity and h is the height of the particle above the bottom of the sphere. As the particle slides down the sphere, its height decreases, so its velocity also decreases. This means that the centripetal acceleration also decreases.

iii. The tangential acceleration of the mass is the acceleration in the direction of motion. In this case, it is equal to the acceleration due to gravity, since the particle is moving downwards. Therefore, the tangential acceleration can be calculated using the formula at = g.

(b) To determine the value of (theta) at which the particle leaves the sphere, we can use conservation of energy. At the top of the sphere, the particle only has potential energy, which can be calculated using the formula PE = mgh. As the particle slides down, it loses potential energy and gains kinetic energy. When the particle leaves the sphere, all of its potential energy has been converted into kinetic energy. Therefore, we can set the equations for potential and kinetic energy equal to each other and solve for (theta). The equation would be:

mgh = 1/2 * m * v^2

Solving for (theta), we get:

(theta) = arccos(sqrt(2gh/R))

This is the angle at which the particle will leave the sphere.