A student wants to determine the coefficients of static...

A student wants to determine the coefficients of static friction and kinetic friction between a box and a plank. She places the box on the plank and gradually raises one end of the plank. When the angle of inclination with the horizontal reaches $28.0^{\circ}$, the box starts to slip and slides $2.53 \mathrm{~m}$ down the plank in $3.92 \mathrm{~s}$. Find the coefficients of friction.

Key Concepts

Kinetic Friction

Kinetic friction is the force that opposes relative movement between surfaces that are already sliding past one another. It is usually lower than static friction and is defined by the coefficient of kinetic friction. In dynamics, this force is crucial for determining the net acceleration of a sliding object, as it works in conjunction with other forces such as the gravitational component along an inclined plane.

Inclined Plane Dynamics

Inclined plane dynamics involves the study of how gravitational force is resolved into components parallel and perpendicular to the surface of a slope. This analysis is essential to understand how the normal force (affecting friction) and the gravitational pull (causing motion) interact. It forms the basis for setting up equations that relate the forces acting on an object on a slope, thus enabling the calculation of friction coefficients.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects by relating displacement, time, velocity, and acceleration without considering the forces causing the motion. In problems involving sliding on an inclined plane, kinematic equations help determine parameters such as acceleration and displacement, which are used in conjunction with frictional forces to analyze the motion and calculate the kinetic friction coefficient.

Static Friction

Static friction is the force that opposes the initiation of motion between two surfaces in contact. It acts counter to the component of gravitational force parallel to the surface and is characterized by the coefficient of static friction, which determines the threshold at which an object begins to slide. This concept is pivotal in understanding how inclined forces overcome frictional resistance at the point of impending motion.

Transcript

00:02 Hello and welcome to this video solution of numerate.

00:06 So here it's given that a student wants to determine the coefficient of static friction and kinetic friction between a box and a plank.

00:15 Right.

00:16 She places the box on a plank and gradually raises one end of the plank.

00:24 Right.

00:25 And when the angle of inclination theta is 28 degrees, the box starts to slip.

00:33 Right.

00:33 And it slides a distance of s let's say which is 2 .53 meters down the plank at a time of 3 .92 seconds.

00:45 Based on this you have to calculate the coefficient of frictions.

00:49 Now this is pretty simple.

00:50 So let's say here you have the block.

00:53 Right.

00:55 And here we see that the weight mg is acting downwards.

01:00 Right.

01:01 A component of this weight that is mg sine theta is acting along the incline.

01:09 We have got mg cos theta acting perpendicular to it.

01:14 The normal reaction is perpendicular to the plane itself.

01:20 And here on the opposite side you will be having the frictional force acting.

01:24 Right.

01:25 Now the box is going to slide if mg sine theta just balances the frictional force.

01:33 Right.

01:33 Then it will slide down.

01:36 Now what you have is mg sine theta will be equal to now the frictional force is nothing but mu times the normal reaction.

01:46 Right.

01:47 And this normal reaction is nothing but mg cos theta.

01:54 So clearly mu or this mu is the coefficient of static friction.

01:58 Mu s is equal to tan theta.

02:02 And this tan of this is 28 degrees.

02:06 This gives you the value of static coefficient of static friction as 0 .53.

02:16 So this is the coefficient of static friction you have got.

02:19 Next to calculate the coefficient of kinetic friction you need to consider the motion of the block.

02:26 Right...

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