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Chapter 12: Problem 6

Is a ladder more likely to slip when you stand near the top or the bottom? Explain.

Short Answer

Expert verified
A ladder is more likely to slip when you are near the top. This is due to the increase in the torque that attempts to rotate the ladder away from the wall as the center of mass is further from the ground. This increased torque can overcome the friction force that prevents the ladder from slipping.

Step by step solution

01

Understanding the Forces Acted on the Ladder

The ladder experiences two types of forces: A downward force due to the weight (gravity) acting at the center of mass, and a horizontal force caused by the person climbing which is exerted at the point of contact with the ground. Additionally, there's a reaction force exerted by the ground on the ladder.
02

Understanding Friction

Friction between the ground and the ladder prevents it from slipping. It's calculated as \( F_{friction} = \mu F_{normal} \), where: \(\mu\) is the coefficient of static friction, \(F_{normal}\) is the normal force exerted by the ladder on the ground. As you climb the ladder, the center of mass rises. Increasing the perpendicular distance of the weight force from the point of contact with the ground increases the torque (rotational force) that attempts to rotate the ladder away from the wall. If this happens, the ladder would slip.
03

Analyzing the Position

When standing at the bottom of the ladder, the center of mass is relatively close to the ground. Thus the torque trying to rotate the ladder around the base and cause a slip, is relatively small. When standing at the top of the ladder, the center of mass is further from the ground, causing an increase in the torque trying to rotate the ladder. If the torque becomes greater than the friction force keeping the ladder in place, the ladder will slip.

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