Vertical Centripetal Forces: Videos & Practice Problems

In vertical circular motion, such as a roller coaster loop, the speed varies due to gravitational force (mg) affecting acceleration. Centripetal acceleration (a_c) is calculated using the equation $v^2/r$. At the bottom, normal force (N_b) is found using $F=ma$, while at the top, it’s adjusted to account for both mg and a_c. Understanding these forces is crucial for analyzing motion in a vertical plane.

Vertical Centripetal Forces

In the study of uniform circular motion, it's essential to understand how centripetal forces operate not only in horizontal planes but also in vertical scenarios, such as roller coasters navigating loops. When analyzing these situations, we apply the same fundamental equations, but we must account for additional forces, particularly gravitational force (mg), which affects the speed of the object throughout its motion.

For a roller coaster cart moving through a vertical loop with a radius (r) of 10 meters, we can determine the centripetal acceleration (a_c) at different points in the loop. At the bottom of the loop, if the speed (v_b) is 30 m/s, we calculate the centripetal acceleration using the formula:

$$ a_c = \frac{v^2}{r} $$

Substituting the values, we find:

$$ a_c = \frac{30^2}{10} = 90 \, \text{m/s}^2 $$

Next, to find the normal force (N_b) acting on the cart at the bottom, we apply Newton's second law, where the net centripetal force is the sum of the forces acting on the cart. At the bottom, the normal force acts upwards while the weight (mg) acts downwards. Thus, we establish the equation:

$$ N_b - mg = m a_c $$

Rearranging gives:

$$ N_b = m a_c + mg $$

Given a mass (m) of 70 kg and gravitational acceleration (g) of 9.8 m/s², we can calculate:

$$ N_b = 70 \times 90 + 70 \times 9.8 = 6300 + 686 = 6986 \, \text{N} $$

At the top of the loop, where the speed (v_t) is 20 m/s, we again calculate the centripetal acceleration:

$$ a_c = \frac{20^2}{10} = 40 \, \text{m/s}^2 $$

For the normal force at the top (N_t), both the normal force and the weight act in the same direction (downwards), leading to the equation:

$$ N_t + mg = m a_c $$

Rearranging gives:

$$ N_t = m a_c - mg $$

Substituting the values, we find:

$$ N_t = 70 \times 40 - 70 \times 9.8 = 2800 - 686 = 2114 \, \text{N} $$

This analysis illustrates that the normal force is significantly greater at the bottom of the loop due to the higher speed and the need to counteract gravitational force, while at the top, the normal force is reduced as the speed decreases. Understanding these dynamics is crucial for solving problems related to vertical circular motion.

Suppose a 1,800-kg car passes over a bump in a roadway that follows the arc of a circle of radius 20m. What
force does the road exert on the car as the car moves over the top of the bump if the car moves at a constant 9 m/s?

• A. 10350 N
• B. 24930 N
• C. 17640 N
• D. 16830 N

Answer: (A)

Here’s what students ask on this topic:

What is centripetal acceleration in vertical circular motion?

Centripetal acceleration in vertical circular motion is the acceleration that keeps an object moving in a circular path. It is directed towards the center of the circle. The formula to calculate centripetal acceleration ($a_c$) is:

$a_c=\frac{v^2}{r}$

where $v$ is the velocity of the object and $r$ is the radius of the circular path. In vertical circular motion, such as a roller coaster loop, the speed varies due to gravitational force ($\mathrm{mg}$) affecting the acceleration.

How do you calculate the normal force at the bottom of a vertical loop?

To calculate the normal force ($N_b$) at the bottom of a vertical loop, you need to consider both the centripetal force and the gravitational force. The formula is:

$N_b=m(a_c+g)$

where $m$ is the mass of the object, $a_c$ is the centripetal acceleration, and $g$ is the acceleration due to gravity (9.8 m/s²). The centripetal acceleration can be calculated using $a_c=\frac{v^2}{r}$, where $v$ is the velocity at the bottom and $r$ is the radius of the loop.

Why does speed vary in vertical circular motion?

In vertical circular motion, the speed varies due to the influence of gravitational force ($\mathrm{mg}$). As an object moves through different points in the vertical loop, gravity either accelerates or decelerates it. At the top of the loop, gravity slows the object down, reducing its speed. Conversely, at the bottom of the loop, gravity accelerates the object, increasing its speed. This variation in speed affects the centripetal acceleration and the forces acting on the object at different points in the loop.

How do you calculate the normal force at the top of a vertical loop?

To calculate the normal force ($N_t$) at the top of a vertical loop, you need to consider both the centripetal force and the gravitational force. The formula is:

$N_t=m(a_c-g)$

where $m$ is the mass of the object, $a_c$ is the centripetal acceleration, and $g$ is the acceleration due to gravity (9.8 m/s²). The centripetal acceleration can be calculated using $a_c=\frac{v^2}{r}$, where $v$ is the velocity at the top and $r$ is the radius of the loop.

What are the key differences between horizontal and vertical circular motion?

The key differences between horizontal and vertical circular motion are:

• Gravitational Force: In vertical circular motion, gravity affects the speed and forces acting on the object, while in horizontal circular motion, gravity does not influence the centripetal force directly.
• Speed Variation: In vertical circular motion, the speed varies due to gravitational force, whereas in horizontal circular motion, the speed is typically constant.
• Force Calculations: In vertical circular motion, you need to account for gravitational force when calculating normal force and centripetal force, while in horizontal circular motion, calculations are simpler as gravity is perpendicular to the motion.