Concerning the astronaut:

The force of gravity will be essentially constant and downward. Thus, in order for acceleration of the astronaut to increase, the normal force must increase.

The free body diagram of the system is shown below:

is the normal force on the block, and is the weight of the block.

Since is a component of , we can represent it as:

If you're confused as why it's cosine and not sine, think about the system practically. The flatter the slope is, the greater the normal force. The smaller an angle becomes (creating a flatter slope), the greater the value of cosine becomes, and subsequently the greater the normal force becomes.

Now we can simply plug in our given values:

To begin solving our problem, we will create a free body diagram, labeling all our forces and the direction in which they are acting.

The normal force, , is the component of force perpendicular to the surface of contact. Weight, , is a force that gravity exerts on an object, acting in the same direction as gravity. To find the normal force, we will begin with Newton's 2nd law:

Where is the sum of all forces in the same direction, is mass, and is the acceleration in the direction of the forces.

Since our object is at rest, and Newton's 2nd law becomes:

Assuming is in the positive direction, the sum of our forces in that direction is

Substituting in the definition of weight, , and solving the expression for gives us

Since and

We need to develop an expression for the normal force on the block to solve this problem:

If you are unsure of whether to use sine or cosine, think about the system practically: the flatter the system gets, the larger the normal force will become; hence the use of the cosine function.

Rearranging for mass, we get:

To begin solving our problem, we will create a free body diagram, labeling all our forces and the direction in which they are acting.

The normal force, , is the component of force perpendicular to the surface of contact. Weight, , is a force that gravity exerts on an object, acting in the same direction as gravity. is the force pressing down on the book.

To find the normal force, we will begin with Newton's 2nd law:

Where is the sum of all forces in the same direction, is mass, and is the acceleration in the direction of the forces.

Since our object is at rest, and Newton's 2nd law becomes

Assuming is in the positive direction, the sum of our forces in that direction is

Substituting in the definition of weight, , and solving the expression for gives us

Since , and

To begin solving our problem, we will create a free body diagram, labeling all our forces and the direction in which they are acting.

The normal force, , is the component of force perpendicular to the surface of contact. Weight, , is a force that gravity exerts on an object, acting in the same direction as gravity.

Weight must be divided into two components, the parts acting parallel and perpendicular to the contact surface.

is the x-component of the weight and is

is the y-component of the weight and is

To find the normal force, we will begin with Newton's 2nd law:

Where is the sum of all forces in the same direction, is mass, and is the acceleration in the direction of the forces.

Since our object is at rest, and Newton's 2nd law becomes

To find , we only need to consider forces acting the same or opposite direction as .

Assuming is in the positive direction, the sum of our forces in that direction is

Substituting in , and solving the expression for gives us

Since , and

There are two forces in play in this scenario. The first is gravity and the second is friction. Both depend on the angle of the slope. Since the block is traveling at a constant rate, we know the the gravitational and frictional force in the direction of the slope cancel each other out, and the net force is zero (there is no acceleration). Therefore, we can write:

Substituting in expressions for each force, we get:

If you are unsure of whether to use cosine or sine for each force, think about the situation practically. The flatter the slope gets, the less the force of gravity will have an effect on moving the block down the plane, hence the use of the sine function. Also, the flatter the slope gets, the greater the normal force will become, hence the use of the cosine function.

Canceling out mass and solving for the angle on one side of the equation, we get:

This is an important property to know! When an object travels down a slope at a constant rate, the tangent of the angle of the slope is equal to the coefficient of kinetic friction.

This question tests your understanding of the concept of normal force. In a block that remains stationary, in which the only two forces acting upon it are the force of gravity, and normal force, you therefore know that the normal force must be equal to the force of gravity, since the block is not accelerating or decelerating (i.e. the sum of the forces is equal to zero).

Thus, we can set the normal force equal to the force of gravity, and solve as follows:

Therefore, the normal force exerted on the block by the ground is .

Using Newton's second law, we can determine the answer.

In our problem, and . is the acceleration due to gravity. Therefore,

The normal force points perpendicular to the force of gravity (opposite direction) and is equal in magnitude. Because the force of gravity is equal to , we simply multiply our given mass and the force of gravity to get our answer.