Question

1. Ball rolling down an incline :Later you will be measuring the acceleration of the rolling ball down the incline for several different incline angles, and we want to compare with the theoretical expected value. The best way to show this is to plot the acceleration on the vertical axis, and the value of sin(θ) on the horizontal axis, as shown on the plot below. Draw in the expected theoretical variation of the acceleration on this plot.

2. A 0.200 kg ball starting from rest rolls down an inclined plane with θ = 30 . Find the acceleration of the ball, and the magnitude of the normal force exerted by the plane.

3. For the ball from question 2, if it starts from rest a height h = 30.0 cm above its position at the bottom of the incline (measured between the center of the ball at each position), what is the speed of the ball v_f at that bottom point, and how long did it take to get there?

4. Ball rolling up the incline : The ball is now set into motion from the bottom of the 30 incline with initial speed v_i = 2.00 m/s. What is the maximum distance d that it travels up the incline before it reverses and travels back? What is its speed arriving back at the bottom, and the total time to go up and down?

Answer 1

The acceleraton of the ball is the horizontal component of the weight of the ball on the inclined plane,

i.e. deducted as

Fh is the horizontal component of the weight of the ball, m is the mass ah is the horizontal acceleration of the ball, g is the gravitatonal constant and is the angle of incline.

Hence,

----------------(1)

the magnitude of normal force (i.e. the vertical component of the weight of the ball) is

--------------(2)

The plot is given as

The plot will be a straight line passing through origin with a slope g (This is deducted from the equation (1)).

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2.

Newtons

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As ball is a sphere its Moment of Inertia is

r is the radius of ball and I is the moment of inertia

Now, the Kinetic Energy is

v is the velocity and is angular velocity and it is equal to v/r

Also, the Potential Energy of the ball is

h is the height at whiich it is

Now, due to the Principle of energy conservation,

The initial potential energy is equal to the final kinetic energy

Hence,

\

So,

note we used angular velocity is equal to v/r.

Find the final velocity using the values given above.

Noe,

Hence,, the time taken for the ball to reach the bottom is

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We again use the principle of conservation of energy,

The initial Kinetic energy will be converted to the potential energy and

Hence,

also note

Hence, using the Moment of Inertia for a sphere we get,

The speed will be the same, as there is no friction.

This is found the procedure as in previous section, the factor of 2 is because it will take same time to go up and come down as there is not friction.