Q: For a block pulled at an angle $$\theta$$ above the horizontal, which equation correctly represents Newton's 2nd Law in the horizontal direction (assuming no friction)?

$$F_{\text{net},x} = T \cos(\theta)$$

Q: A block is pulled at an angle $$\theta$$ above the horizontal with force $$T. If $$the coefficient of kinetic friction is $$\mu_k$$, what is the expression for the frictional force?

$$f_k = \mu_k (mg - T \sin(\theta))$$

Q: A cart is pushed up a ramp with initial speed $$v_0. If $$the ramp angle is $$\theta$$ and the coefficient of kinetic friction is $$\mu_k$$, what is the acceleration of the cart along the ramp?

$$a = -g \sin(\theta) - \mu_k g \cos(\theta)$$

Q: A block of mass $$m is $$pulled at an angle $$\theta$$ above the horizontal with force $$T. If $$the block moves at constant velocity, what is the relationship between the pulling force and friction?

$$T \cos(\theta) = \mu_k (mg - T \sin(\theta))$$

Q: A cart is pushed up a ramp and comes to rest. If the coefficient of static friction is $$\mu_s$$, what condition must be met for the cart to remain at rest?

$$mg \sin(\theta) \leq \mu_s mg \cos(\theta)$$

Q: A block is pulled at an angle $$\theta$$ above the horizontal. If the vertical component of the pulling force is $$T \sin(\theta)$$, how does this affect the normal force?

The normal force decreases by $$T \sin(\theta).$$

Q: A cart is pushed up a ramp with initial velocity $$v_0. If $$the ramp angle is $$\theta$$ and friction is present, which equation describes the net force along the ramp?

$$F_{\text{net}} = -mg \sin(\theta) - \mu_k mg \cos(\theta)$$

Question 1

A pendulum inside a car hangs at an angle $$	heta = 40.9^\circ$$ from the vertical, as shown in the first image. What does this angle indicate about the motion of the car?

Accepted Answer

The car is accelerating forward, and the angle indicates the direction of the net force.

Question 2

A block of mass $$0.7\ 	ext{kg} is $$pulled with a force of $$41\ 	ext{N} at an $$angle of $$50^\circ$$ above the horizontal, as shown in the second image. What is the horizontal component of the pulling force?

Accepted Answer

$$41 	imes \cos(50^\circ)$$

Question 3

A block of mass $$1216\ 	ext{g}$$ slides up a ramp at $$3.443\ 	ext{m/s} ($$third image). If the ramp angle is $$	heta$$, what is the component of the gravitational force acting down the ramp?

Accepted Answer

$$mg \sin(	heta)$$

Question 4

A cart of mass $$856\ 	ext{g}$$ moves up a ramp at $$1.253\ 	ext{m/s} ($$fourth image). If friction is negligible, what is the net force acting on the cart along the ramp?

Accepted Answer

$$-mg \sin(	heta)$$

Question 5

For a block pulled at an angle $$	heta$$ above the horizontal, which equation correctly represents Newton's 2nd Law in the horizontal direction (assuming no friction)?

Accepted Answer

$$F_{	ext{net},x} = T \cos(	heta)$$

Question 6

A block slides up a ramp with initial velocity $$v_0. If $$the ramp angle is $$	heta$$ and friction is ignored, what is the acceleration of the block along the ramp?

Accepted Answer

$$a = -g \sin(	heta)$$

Question 7

A block of mass $$m$$ rests on a ramp inclined at angle $$	heta ($$seventh image). What is the normal force acting on the block?

Accepted Answer

$$n = mg \cos(	heta)$$

Question 8

A block is pulled at an angle $$	heta$$ above the horizontal with force $$T. If $$the coefficient of kinetic friction is $$\mu_k$$, what is the expression for the frictional force?

Accepted Answer

$$f_k = \mu_k (mg - T \sin(	heta))$$

Question 9

A cart is pushed up a ramp with initial speed $$v_0. If $$the ramp angle is $$	heta$$ and the coefficient of kinetic friction is $$\mu_k$$, what is the acceleration of the cart along the ramp?

Accepted Answer

$$a = -g \sin(	heta) - \mu_k g \cos(	heta)$$

Question 10

A block of mass $$m is $$pulled at an angle $$	heta$$ above the horizontal with force $$T. If $$the block moves at constant velocity, what is the relationship between the pulling force and friction?

Accepted Answer

$$T \cos(	heta) = \mu_k (mg - T \sin(	heta))$$

Question 11

A block slides up a ramp and comes to a stop after traveling distance $$d. If $$the initial velocity is $$v_0$$, ramp angle is $$	heta$$, and acceleration is $$a$$, what equation gives the distance traveled?

Accepted Answer

$$d = \frac{v_0$^2}{2|a|}$$

Question 12

A block is placed on a ramp inclined at angle $$	heta. $$Which component of the weight acts perpendicular to the ramp?

Accepted Answer

$$mg \cos(	heta)$$

Question 13

A cart is pushed up a ramp and comes to rest. If the coefficient of static friction is $$\mu_s$$, what condition must be met for the cart to remain at rest?

Accepted Answer

$$mg \sin(	heta) \leq \mu_s mg \cos(	heta)$$

Question 14

When choosing axes for analyzing forces on a ramp, why is it advantageous to align one axis along the ramp?

Accepted Answer

It simplifies calculations by making one direction sum to zero and the other direction sum to the net force.

Question 15

A block is pulled at an angle $$	heta$$ above the horizontal. If the vertical component of the pulling force is $$T \sin(	heta)$$, how does this affect the normal force?

Accepted Answer

The normal force decreases by $$T \sin(	heta).$$

Question 16

A cart is pushed up a ramp with initial velocity $$v_0. If $$the ramp angle is $$	heta$$ and friction is present, which equation describes the net force along the ramp?

Accepted Answer

$$F_{	ext{net}} = -mg \sin(	heta) - \mu_k mg \cos(	heta)$$

Question 17

A block is pulled at an angle $$	heta$$ above the horizontal. If the block is accelerating, which equation represents Newton's 2nd Law in the horizontal direction?

Accepted Answer

$$ma = T \cos(	heta) - f_k$$

Question 18

A block slides up a ramp and comes to a stop. If the initial velocity is $$v_0$$, ramp angle is $$	heta$$, and acceleration is $$a$$, what is the time taken to stop?

Accepted Answer

$$t = \frac{v_0$}{|a|}$$

EXAM #3 DAY 15: Pulling at Angles and Forces on Ramps: Newton’s Laws in 2D

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