Question

write at least 5 sentences answering the question, "How does the height of a ramp affect the speed of an object rolling down that ramp?"

Answer 1

Newton's second law allows us to find the effect of the height of the ramp on the movement of the block.

• When the height increases the acceleration of the body increases

• Tthere is a minimum height for which the block does not slide down the ramp

Newton's second law says that force is proportional to the product of mass and the acceleration of bodies

                F = m a

Where F is the net force, m and a are the mass and acceleration of the body, the bold letters indicate vectors.

To use this expression we must set a reference frame and apply Newton's second law to each axis.

The free body diagram is a method to visualize the forces in a problem without the details of the shape of the objects, in the attached we can see a free body diagram, the x axis is placed parallel to the ramp with the positive direction in the direction of motion and the y-axis in the vertical direction

Let's write Newton's second law for each axis

y-axis

            $N - W_y$ = 0

            N = mg cos tea

Let's write the expression for the force

x-axis

                 $W_x - fr = ma$

                 mg sin θ - fr = m a

The friction force is the macroscopic sum of all the microscopic forces between the two surfaces and can be written

                 fr = μ N

                 fr = μ mg cos θ

We substitute

              a = g (sin θ - μ cos θ

This expression gives the form of the block's acceleration as it descends the ramp

The height of the ramp can be found with trigonometry

            sin θ = h / L

            h = L sin θ

Let's write our equations

            a = g ( sin θ - μ cos θ)

            h = L sin θ

We substitute

            a = g (h / L - μ cos θ )

Let's analyze this expression;

It is observed that the acceleration increases with the height of the ramp

When the height increases, the angle of the ramp must increase, therefore the second term, which is a cosine function, decreases, so the acceleration also increases.

The angle decreases so much the height of the ramp decreases with the second term it increases to the point where the acceleration becomes zero

Consequently, using Newton's second law we can find the effect of the height of the ramp.

• When the height increases the acceleration of the body increases

• There is a minimum height for which the block does not slide down the ramp

Learn more about Newton's second law here:

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