Tension - AP Physics 1

Since the gravitational force must be cancelled by the tension force, as the ball is experiencing no acceleration, and no other forces are being applied to it:

Luckily enough, the angles of the two ropes are the same. Therefore, the tension in each will be the same. This immediately eliminates two of the five answers. Now we just need to calculate what that force is.

We know that together, the vertical components of the tension must equal the weight of the block. Therefore we can write:

Since we know that the two tension forces are equal, we can rewrite:

Rearranging for T, we get:

Since there is no friction between the mass and slope, there are only two relevant forces acting on the mass: gravity and tension. Furthermore, since the block is not in motion, we know that these forces are equal to each other. Therefore:

Substituting in an expression for the force of gravity, we get:

We know all of these values, allowing us to solve for the tension:

There are three relevant forces acting on the block in this scenario: friction, tension, and gravity. We are given two of these values, so we simply need to develop an expression for the force of gravity in the direction of the slope. Since the block is motionless, we can write:

Substituting in an expression for the force of gravity, we get:

Rearrange to solve for the angle:

We know all of these values, allowing us to solve:

There are three relevant forces acting on the block in this scenario: tension, friction, and gravity. We are given tension, so we will need to develop expressions for friction and gravity. Since the block is motionless, we can say:

Plugging in expressions for the force of gravity and friction, we get:

Rearranging for the mass, we get:

We know all of these values, allowing us to solve:

Since there is no tension, there are only two relevant forces acting on the block: friction and gravity. Since the block is motionless, we can also write:

Substitute the expressions for these two forces:

Canceling out mass and gravitational acceleration, and rearranging for the coefficient of static friction, we get:

There are three relevant forces acting on the block in this situtation: friction, gravity, and tension. We can use Newton's second law to express the system:

Substituting expressions in for the forces, we get:

Canceling out mass and rearranging to solve for tension, we get:

We have values for each variable, allowing us to solve:

The block has three forces on it: the force of tension, the force of gravity, and the force from Bruce. The force of gravity is:

The force from Bruce plus the force of tension has to equal gravity (since Bruce's force and tension are up while gravity is down) so the block is in equilibrium.

In order to find the mass of block 2, we're going to need to calculate a few other things, such as the tension in the rope.

To begin with, we'll need to identify the various forces on our free-body diagram. To do this, we will begin with block 1 and use a rotated coordinate system to simplify things. In such a system, the x-axis will run parallel to the surface of the ramp, while the y-axis will be perpendicular to the ramp's surface, as shown below:

Now we can identify the forces acting on block 1. Along the rotated y-axis, the force of gravity acting on the block is equal to , and the force of the ramp on the block is just the normal force, . Since block 1 is not moving in the y direction, we can set these two forces equal to each other.

Now, considering the forces acting along the rotated x-axis, we have a force pointing downwards equal to . Pointing upwards, we have the tension force and we also have the frictional force, .

The formula for calculating the force due to kinetic friction is:

Since we have already determined what the normal force is, we can substitute that expression into the above equation to obtain:

Now, we can write an expression for the net force acting upon block 1 in the x direction:

Rearrange the above expression to solve for tension.

So far, we have only been looking at block 1. Now let's turn our attention to block 2 and see what forces are acting on it. In the downward direction we have the weight of the block due to gravity, which is equal to . In the upward direction, as we can see in the diagram, we have the tension of the rope, . We need to write an expression that tells us the net force acting upon block 2.

Since we calculated the expression for tension from the information regarding block 1, we can plug that expression into the above equation in order to obtain:

Now rearrange to solve for the mass of block 2.

Then plugging in values, we can finally calculate block 2's mass:

Here is a (simplistic) diagram of the elevator. There are three forces acting on this object. There is it's own weight due to gravity ; the tension in the rope holding it up, which we will call ; and there is an external force due to the fact that the elevator is accelerating upward. Since the mass is fixed, it has zero velocity with respect to the elevator, and therefore the net force on the mass is 0 by Newton's 2nd law. The net force is given by . Solving this equation for and plugging in the values gives us:

Since we want the magnitude of the tension, and not the value of the vector, we omit the minus sign. Therefore the tension in the rope is , as desired.

Plug in values:

Solve for

The velocity of the earth is

Convert to :

Plug in values:

For this question, we're presented with a scenario in which an object of a given mass is hanging to a rope. That rope is being pulled on, causing the object to accelerate upwards at a certain rate. We're then asked to calculate the tension that results in the rope.

In order to answer this question, it's best if we approach this by examining a force diagram. Since there are no pertinent forces occurring in the x direction, the only concern we have is in the y direction.

One of the forces acting on the object is the downward force of gravity. Another force is the upward tension caused by the rope. Hence, we know what force components contribute to the net force. Furthermore, we're told that as it is pulled up, the object accelerates at a rate of . Since this is the net acceleration in the y direction, we can determine the net force in the y direction.

Next, we can rearrange the above terms in order to isolate the term for tension.

Finally, if we plug in the values that we know, we can calculate our answer.

The masses will each provide an equal force to each other through the rope. Thus, there will be no net force and no net acceleration.

Using superposition of forces and definition of force:

All four ropes will be providing the same force.

Combining equations

Solving for , the force of tension in one rope:

Plugging in values, remembering that gravity is pointing down and thus will be a negative number.

To answer this question, we need to separately consider the component forces acting in the horizontal and vertical direction.

First, let's consider the vertical component. Since the box is only moving in the horizontal direction, we know that there are no net forces acting in the vertical direction. Consequently, the net force in the y-direction is zero.

Next, let's look at the forces acting in the horizontal direction. Since we need to figure out the force in the string necessary to make the box move at a constant speed, we're looking for a situation in which the box is not accelerating in the horizontal direction. Thus, we're looking for a case in which the net force in the x-direction is zero.

Now, let's rewrite the expression for the force of kinetic friction.

Now, plugging in the expression for the normal force, we obtain the following.

Next, let's go ahead and plug in the values given to us in the question stem.

The swinging back and forth motion is a part of a circle, thus

Combining equations

There will be the force of gravity and the string acting on the coin

Solving for tension:

Converting and plugging in values: