Prove that tanø is equal to U(k) when the block slides down the incline with a constant speed. (Use symbols, not numbers.)
To prove that tanø is equal to μk when a block slides down an incline with a constant speed, we can use the concepts of friction and equilibrium. Here's how to approach the problem step by step:
Step 1: Draw a free-body diagram of the block. Identify all the forces acting on the block. In this case, there are two main forces: the force of gravity acting downward (mg) and the force of friction opposing motion along the incline (fk).
Step 2: Break the force of gravity (mg) into two components: one parallel to the incline (mg∥) and one perpendicular to the incline (mg⊥). The angle ø is the angle between the incline and the horizontal direction.
Step 3: The force of friction (fk) can be represented as μk times the normal force (N) exerted by the incline on the block. The normal force (N) is equal to mg⊥.
Step 4: In order for the block to slide down the incline with a constant speed (no acceleration), we know that the net force along the incline must be equal to zero.
Step 5: Write the equation for the net force along the incline:
Net force along incline = m * acceleration along incline
Since the block has a constant speed, the acceleration along the incline is zero.
Net force along incline = 0
Step 6: Use trigonometry to represent the forces in terms of ø:
Net force along incline = mg∥ - fk
Net force along incline = mg*sinø - (μk * N)
Since mg∥ = mg*sinø and N = mg⊥ = mg*cosø, we can substitute the expressions:
0 = mg*sinø - (μk * mg*cosø)
Step 7: Simplify the equation:
0 = mg(sinø - (μk*cosø))
Step 8: Divide both sides of the equation by mg:
sinø - (μk*cosø) = 0
Step 9: Add (μk*cosø) to both sides of the equation:
sinø = (μk*cosø)
Step 10: Use the identity tanø = sinø / cosø:
tanø = (μk*cosø) / cosø
Step 11: Simplify the equation:
tanø = μk
Therefore, we have proved that tanø is equal to μk when the block slides down the incline with a constant speed.
To prove that tanø is equal to U(k) when the block slides down the incline with a constant speed, we can follow these steps:
Step 1: Identify the given information:
- The block is sliding down the incline with a constant speed.
- We need to prove that tanø is equal to U(k).
Step 2: Define the symbols:
- Let ø represent the angle of inclination of the incline.
- Let U(k) represent the coefficient of kinetic friction.
Step 3: Determine the forces acting on the block:
- The force of gravity acting vertically downward can be represented by mg, where m is the mass of the block and g is the acceleration due to gravity.
- The normal force acting perpendicular to the incline can be represented by N.
- The frictional force opposing the motion can be represented by f(k).
Step 4: Resolve the force of gravity into components:
- The component of the force of gravity parallel to the incline can be represented by mg*sin(ø), where sin(ø) is the sine of the angle of inclination ø.
- The component of the force of gravity perpendicular to the incline is balanced by the normal force, which can be represented by N = mg*cos(ø), where cos(ø) is the cosine of the angle of inclination ø.
Step 5: Determine the net force acting on the block:
- The net force parallel to the incline is the difference between the force of gravity parallel to the incline and the frictional force: f(net) = mg*sin(ø) - f(k).
Step 6: Set up the equation for the constant speed condition:
- Since the block is sliding down the incline with a constant speed, the net force parallel to the incline must be zero: f(net) = 0.
Step 7: Substitute the equation for the net force into the constant speed condition equation:
mg*sin(ø) - f(k) = 0
Step 8: Solve the equation for the coefficient of kinetic friction, U(k):
f(k) = mg*sin(ø)
f(k) = U(k)*N
mg*sin(ø) = U(k)*mg*cos(ø)
Step 9: Simplify the equation:
tan(ø) = U(k)
Step 10: Finally, we have proved that tanø is equal to U(k) when the block slides down the incline with a constant speed.
Please note that this proof assumes that there are no other external forces acting on the block besides gravity and friction, and that the incline is frictional.