Acceleration of a ball down an incline?

Question

A spherical ball is placed on a plane surface inclined
at θ to the horizontal. Determine the linear
acceleration of the center of the sphere down the
incline if:
1) θ = 40 degrees
2) θ = 50 degrees

Coefficient of static friction, μs = 0.3
Coefficient of kinetic friciton, μk = 0.2
Mass moment of inertia of a sphere, J = 2/5 m*R^2

janbbeck: No it doesn't. Work it out and you'll see that the m always cancels out.

doug: smoking is not good for you, although coffee is very good.

civil: the fact that it is rolling means that you cannot ignore friction - static or kinetic.

Note also that the ball could be rolling, sliding, or BOTH.

The actual value of the frictional force depends on whether the ball is rolling without sliding or rolling with sliding.

If sliding, f = μk*N
If not sliding f <= μs*N

The angle θ determines whether or not the ball slides.

Answer 1

from a FBD of the ball on the plane and rotating the reference frame along the plane we can conclude the following

sum of the forces normal to the plane = 0 = N - mg*cos(θ)
N = mg*cos(θ)
where N is the normal force

****
taking static friction into account
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sum of the forces parallel to the plane = m*a = mg*sin(θ) - μk*N - μs*N = mg*sin(θ) - μk(mg*cos(θ)) - μs(mg*cos(θ)
a = g*sin(θ) - μk*g*cos(θ) - μs*g*cos(θ)

****
the above formulation takes care of it. if the ball is sliding then μs = 0. if it is not sliding then μk = 0. we're assuming that the ball is on a point load and the point load cannot have both motions a 1-D problem.

Answer 2

The force acting on the sphere (assuming 1 standard G of 9.8 m/s²) is 9.8 sin(Φ).
Doing the linear to rotational momentum transfer and subtracting it from the total force is --way-- more than I want to get into until I've had my morning coffee and cigarettes ☺

Doug

Answer 3

while a ball rolls down an susceptible plane, it valuable properties speed using fact of gravity. while rolling up, it loses speed using fact of gravity. Why would not gravity play a function while it rolls on a horizontal floor? Gravity would not reason the ball to learn or lose speed. the load of the ball is the rigidity that motives the ball to improve up because it strikes downward. The equation for weight is shown under: Weight = mass * g g = 9.8 m/s^2 The letter g, which a lot of human beings call gravity, skill the load motives the mass to improve up right away down in direction of THE EARTH at 9.8 m/s each 2nd. while a ball is on an susceptible plane, it may no longer improve up right away down, using fact the ball is sitting on suitable of the exterior of the susceptible plane. The ball can in user-friendly terms roll interior the direction parallel to the exterior of the susceptible plane. The equation under describes the portion of the load that motives the ball to improve up because it rolls down the susceptible plane. rigidity parallel = Weight * sin ?, ? is the attitude that the plane has been circled above horizontal. This rigidity continually pulls the ball DOWN the susceptible plane. So the fee of a ball rolling DOWN the plane will improve, and the fee of a ball rolling UP the susceptible plane will decrease. while a ball is rolling on a horizontal floor, it may no longer improve up right away down, using fact the ball is sitting on suitable of the horizontal floor. in spite of the undeniable fact that, the weigh of the ball pushes DOWN on the tough horizontal floor inflicting friction rigidity. Friction rigidity DECELERATES the ball because it rolls on a horizontal floor. Weight continually has some effect on the fee of a shifting merchandise.

Answer 4

Does this not invalidate the answer to the question this is the follow up to? Since the moment of inertia is not independent of mass?
http://answers.yahoo.com/question/index?qid=20070910053630AAU2Qn1&r=w#NbUvWDK_Ujhj0TG8mX3H
sorry I could not contact any other way.