A block is pushed across a floor by a constant force that is...

A block is pushed across a floor by a constant force that is applied at downward angle $\theta($ Fig. $6-19) .$ Figure $6-36$ gives the acceleration magnitude $a$ versus a range of values for the coefficient of kinetic friction $\mu_{k}$ between block and floor: $a_{1}=3.0 \mathrm{m} / \mathrm{s}^{2}, \mu_{k 2}=$ $0.20,$ and $\mu_{k 3}=0.40 .$ What is the value of $\theta ?$

A block is pushed across a floor by a constant force that is applied at downward angle $\theta($ Fig. $6-19) .$ Figure $6-36$ gives the acceleration magnitude $a$ versus a range of values for the coefficient of
kinetic friction $\mu_{k}$ between block and floor: $a_{1}=3.0 \mathrm{m} / \mathrm{s}^{2}, \mu_{k 2}=$
$0.20,$ and $\mu_{k 3}=0.40 .$ What is the value of $\theta ?$

Key Concepts

Force Decomposition

This concept involves breaking down a force applied at an angle into its horizontal and vertical components. In dynamics, especially with forces applied at an incline or at an angle, understanding these components is crucial because the horizontal component drives acceleration while the vertical component affects the normal force acting on the object.

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and the acceleration it undergoes (F = ma). This principle is instrumental in analyzing the horizontal motion of the block where the net force, after accounting for friction and other forces, determines the block's acceleration.

Kinetic Friction

Kinetic friction is the force that resists the motion of two surfaces sliding past each other, calculated as the product of the coefficient of kinetic friction and the normal force. In problems involving motion on surfaces, recognizing how kinetic friction acts and how it is influenced by forces that alter the normal force is essential for determining net acceleration.

Normal Force Modification

When a force is applied at an angle, the vertical component of that force modifies the normal force exerted by the surface on the object. An increase in normal force leads to an increase in the frictional force, which in turn impacts the net force available for acceleration. Understanding this modification is key in solving dynamics problems that involve inclined or angled forces.

Transcript

00:01 So let's draw first the free body diagram for the system.

00:05 We have a point mass.

00:07 We'll treat it as a point mass.

00:09 Going up is force normal.

00:12 Going down is the weight m .g.

00:15 At an angle, we have f.

00:20 Making an angle with the horizontal theta.

00:23 And opposite to the direction of motion would be the force of friction.

00:27 At this point, we can apply the sum of forces in the x direction.

00:30 This will equal the mass times the acceleration.

00:34 This would equal f cosine of theta minus the force of friction.

00:42 We can say the sum of forces in the y direction would be equal to force normal minus the force sine of theta minus mg.

00:53 This is equaling zero because there isn't any acceleration in the y direction.

00:58 We know that here the force of friction would be equal to the code.

01:03 Of kinetic friction times the force normal and according to this equation this will be equal to the coefficient of the coefficient of friction the kinetic coefficient of friction multiplied by m g plus f sign of theta so at this point we can then substitute this into the first equation and say f cosine of theta minus the coefficient of kinetic friction times m g minus f sign of theta would be equal to the mass times the acceleration.

01:39 And so we can solve for the acceleration.

01:41 This is equaling f over m cosine of theta minus the coefficient of kinetic friction times sine of theta.

01:50 This is equaling, rather minus the coefficient of kinetic friction times g...

Please give Ace some feedback