A block of mass 2.30 kg slides down a slope at an angle of 33° to the horizontal, with a constant velocity of 1.0 m s–1. Calculate the coefficient of dynamic friction between the slope and the block and express your answer to two significant figures.

To solve this problem, we need to use the principles of physics, specifically Newton's second law and the equations for friction. Here are the steps:

1. Identify the forces acting on the block. These are gravity, the normal force from the slope, and friction.

2. Resolve the gravitational force into components parallel and perpendicular to the slope. The component of gravity acting down the slope is given by mgsin(θ), where m is the mass of the block, g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of the slope. The component of gravity acting perpendicular to the slope is mgcos(θ).

3. The normal force is equal to the component of gravity acting perpendicular to the slope, so N = mg*cos(θ).

4. The frictional force is given by f = μN, where μ is the coefficient of friction we're trying to find. Since the block is moving at a constant velocity, the frictional force must be equal to the component of gravity acting down the slope, so we have f = mg*sin(θ).

5. Set these two expressions for the frictional force equal to each other and solve for μ:

   μN = mg*sin(θ)

   μmgcos(θ) = mgsin(θ)

   μ = tan(θ)

6. Substitute the given angle into this equation to find μ. Remember to convert the angle to radians first if your calculator is set to radian mode:

   μ = tan(33°) = 0.65

So, the coefficient of dynamic friction between the slope and the block is 0.65 to two significant figures.

To solve this problem, we need to understand that the block is moving at a constant velocity. This means that the forces acting on the block are balanced.

Step 1: Identify the forces acting on the block There are three forces acting on the block:

• The weight of the block (W), which acts vertically downwards. This can be calculated using the formula W = mg, where m is the mass of the block and g is the acceleration due to gravity.
• The normal force (N), which acts perpendicular to the slope.
• The frictional force (F), which acts parallel to the slope and opposes the motion of the block.

Step 2: Resolve the weight into components The weight of the block can be resolved into two components: one acting perpendicular to the slope (Wcosθ) and one acting parallel to the slope (Wsinθ).

Step 3: Set up the equations of motion Since the block is moving at a constant velocity, the net force acting on it is zero. This gives us two equations:

• In the direction perpendicular to the slope: N = Wcosθ
• In the direction parallel to the slope: F = Wsinθ

Step 4: Calculate the frictional force The frictional force can be calculated using the equation F = μN, where μ is the coefficient of friction. Substituting the equation for N from step 3 gives F = μWcosθ.

Step 5: Equate the two expressions for the frictional force Setting the two expressions for the frictional force equal to each other gives μWcosθ = Wsinθ.

Step 6: Solve for the coefficient of friction Solving the equation from step 5 for μ gives μ = sinθ/cosθ = tanθ.

Substituting the given angle of 33° into this equation gives μ = tan(33°) = 0.65 to two significant figures.

To solve this problem, we need to use the equations of motion and the definition of the coefficient of friction. Here are the steps:

1. First, we need to calculate the force of gravity acting on the block. This is given by F = mg, where m is the mass of the block and g is the acceleration due to gravity. So, F = 2.30 kg * 9.8 m/s^2 = 22.54 N.

2. The force of gravity can be split into two components: one acting perpendicular to the slope (F_perpendicular = F * cos(33°)) and one acting parallel to the slope (F_parallel = F * sin(33°)).

3. The force acting perpendicular to the slope is F_perpendicular = 22.54 N * cos(33°) = 18.91 N.

4. The force acting parallel to the slope is F_parallel = 22.54 N * sin(33°) = 12.32 N.

5. Since the block is moving at a constant velocity, the net force acting on it is zero. This means the force of friction acting on the block (F_friction) must be equal to the force acting parallel to the slope (F_parallel). So, F_friction = 12.32 N.

6. The force of friction is also given by F_friction = μN, where μ is the coefficient of friction and N is the normal force. The normal force is equal to the force acting perpendicular to the slope (F_perpendicular). So, μ = F_friction / N = 12.32 N / 18.91 N = 0.65.

So, the coefficient of dynamic friction between the slope and the block is 0.65 to two significant figures.